compact high-order finite difference schemes for acoustic
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Compact High-order Finite Difference Schemes for
Acoustic Wave Equations
Li, Keran
Li, K. (2021). Compact High-order Finite Difference Schemes for Acoustic Wave Equations
(Unpublished doctoral thesis). University of Calgary, Calgary, AB.
http://hdl.handle.net/1880/112939
doctoral thesis
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UNIVERSITY OF CALGARY
Compact High-order Finite Difference Schemes for Acoustic Wave Equations
by
Keran Li
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
GRADUATE PROGRAM IN MATHEMATICS AND STATISTICS
CALGARY, ALBERTA
JANUARY, 2021
c© Keran Li 2021
Abstract
This study developed three compact high-order finite difference schemes for acoustic
wave equations. Benefiting from the compactness, the new schemes require less layers of
boundary conditions than conventional finite difference schemes. All the three schemes
work for acoustic wave equations with variable coefficients in homogeneous media, with
the third one also being applicable to the case of heterogeneous density media. The first
scheme is based on Pade approximation which is formally a product of the inverse of a
finite difference operator and the conventional 2nd-order finite difference operator, thus some
algebraic manipulation is necessary to discuss the product of operators. The second scheme
is based on so-called combined finite difference method, which needs the boundary conditions
for the second spatial derivatives and the needed boundary conditions can be derived by using
the wave equation and usual Dirichlet boundary conditions themselves. The third scheme
is also based on combined finite difference method, and it generalizes the second scheme so
that it can also work in heterogeneous density media case, i.e., the Laplacian in the wave
equations being divergence form. The stability of the first two schemes are established by
an energy method, while the stability of the last scheme is obtained by an analogy of von
Neumann analysis. All of these new schemes are proven to be conditionally stable with
given Courant-Friedrichs-Lewy (CFL) numbers. Numerical experiments are conducted to
verify the efficiency, accuracy and stability of the new schemes. It is expected that these
new schemes will find extensive applications in both research and engineering areas.
Keywords: Acoustic Wave Equation, Compact Finite Difference Scheme, High-order Scheme.
ii
Acknowledgements
I would like to express my sincere gratitude to my supervisor Prof. Wenyuan Liao for his
consistent guidance and encouragement throughout this study. This study would not have
been possible without the invaluable support from Prof. Liao. I am also grateful for the
advice from the other committee members: Prof. Michael Lamoureux, Prof. Antony Ware,
Prof. Elena Braverman and Prof. Dong Liang.
Furthermore I would like to thank my family and my friends who have always been
backing me unconditionally.
I am thankful to the Department of Mathematics and Statistics and all its staff members
for all their help.
iii
To My Family
iv
Table of Contents
Abstract ii
Acknowledgements iii
Dedication iv
Table of Contents v
List of Figures and Illustrations vii
List of Tables viii
List of Symbols, Abbreviations and Nomenclature ix
1 Introduction 1
2 A Compact High Order Alternating Direction Implicit Method for Three-dimensional Acoustic Wave Equations with Variable Coefficients 52.1 Acoustic Wave Equation and Pade Approximation . . . . . . . . . . . . . . . 52.2 Derivation of the Compact High-order ADI Method . . . . . . . . . . . . . . 92.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 An Efficient and High Accuracy Finite Difference Scheme for AcousticWave Equations 323.1 The Compact High-order Scheme . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3 Higher-Order Temporal Accuracy . . . . . . . . . . . . . . . . . . . . . . . . 453.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.4.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
v
3.4.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 Efficient and Stable Finite Difference Modeling of Acoustic Wave Propa-gation in Variable Density Media 584.1 The New Compact Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2 Stability Analysis of the New Scheme . . . . . . . . . . . . . . . . . . . . . . 63
4.2.1 Estimation of the Eigenvalues . . . . . . . . . . . . . . . . . . . . . . 664.2.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.3.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5 Conclusion 83
A Appendix for Chapter 2 85A.1 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
B Appendix for Chapter 3 89B.1 Details of the Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89B.2 Example on Boundary Values in Runge-Kutta Methods . . . . . . . . . . . . 92
C Appendix for Chapter 4 95C.1 PML Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Bibliography 97
vi
List of Figures and Illustrations
2.1 Snapshots of x-section of wavefields computed by the new 4th-order compactmethod at (a) t = 0.2s; (b) t = 0.3s; (c) t = 0.4s; (d) t = 0.5s. . . . . . . . . 29
2.2 Snapshots of y-section of wavefields computed by the new 4th-order compactmethod at (a) t = 0.2s; (b) t = 0.3s; (c) t = 0.4s; (d) t = 0.5s. . . . . . . . . 30
2.3 Snapshots of z-section of wavefields computed by the new 4th-order compactmethod at (a) t = 0.2s; (b) t = 0.3s; (c) t = 0.4s; (d) t = 0.5s. . . . . . . . . 31
3.1 Underground model for Example 3.4.4, the yellow part is soil with sound speedν = 1200m/s, the gray part is rock with sound speed ν = 2500m/s. The wavegenerator is located in soil area. . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Snapshot of y-section at t = 0.225s in Example 3.4.4 . . . . . . . . . . . . . 543.3 Snapshot of y-section at t = 0.375s in Example 3.4.4 . . . . . . . . . . . . . 553.4 Snapshot of y-section at t = 0.42s in Example 3.4.4 . . . . . . . . . . . . . . 563.5 Snapshot of y-section at t = 0.66s in Example 3.4.4 . . . . . . . . . . . . . . 57
4.1 Snapshot of y-section of the acoustic pressure u at t = 0.4s in Example 4.3.2 734.2 Snapshot of y-section of the acoustic pressure u at t = 0.9s in Example 4.3.2 744.3 Snapshot of y-section of the acoustic pressure u at t = 1.4s in Example 4.3.2 754.4 Snapshot of z-section of the acoustic pressure u at t = 0.4s in Example 4.3.2 764.5 Snapshot of z-section of the acoustic pressure u at t = 0.9s in Example 4.3.2 774.6 Snapshot of z-section of the acoustic pressure u at t = 1.4s in Example 4.3.2 784.7 Simulation of Marmousi-2 model. (a): acoustic velocity of the Marmousi-2
model. (b): density of the Marmousi-2 model. . . . . . . . . . . . . . . . . . 794.8 Computation domain with PML zone: Ω = Ω0∪ΩPML, where Ω0 is the original
domain and ΩPML is the PML domain with width 800m . . . . . . . . . . . . 804.9 Snapshots of the acoustic pressure u at different time. . . . . . . . . . . . . . 814.10 Acoustic energy of the wavefield. . . . . . . . . . . . . . . . . . . . . . . . . 82
vii
List of Tables
2.1 Numerical errors in maximal norm for Example 2.4.1 with τ = 0.0025 at T = 1. 242.2 Numerical errors in maximal norm for Example 2.4.1 with various h and τ . 242.3 Numerical errors in maximal norm for Example 2.4.2 with various h and τ . 252.4 Numerical errors in maximal norm for Example 2.4.2 with τ
h< 1
2√
3. . . . . . 26
2.5 Numerical errors in maximal norm for Example 2.4.2 with τh> 1
2√
3. . . . . . 27
3.1 Numerical errors in max norm and energy norm in Example 3.4.1 with τ = h2.Note that the CFL condition requires τ
h<√
318
. . . . . . . . . . . . . . . . . . 483.2 Numerical errors in max norm and energy norm in Example 3.4.2 with τ = h2.
Note that the CFL condition requires τh<√
69
. . . . . . . . . . . . . . . . . . 493.3 Comparison of RE and RK4 in Example 3.4.3 in energy norm, with τ = h
10. . 51
3.4 Comparison of the reference solution and numerical solutions in Example 3.4.4with τ = 0.5× 10−4s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1 Numerical errors in max norm in Example 4.3.1 with τ = h2. Note that theCFL condition requires τ
h< 0.1156. . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Numerical errors in Example (4.3.3) in max norm. Here τ =(
5hπ
)2. . . . . . 76
viii
List of Symbols, Abbreviations andNomenclature
Symbol or abbreviation Definition
∂ The boundary of∂t Partial differential operator with respect to t∂x Partial differential operator with respect to x∂y Partial differential operator with respect to y∂z Partial differential operator with respect to z∆ Laplacian of∇ Gradient of∇· Divergence ofδt Temporal difference operatorδx Spatial difference operator with respect to xδy Spatial difference operator with respect to xδz Spatial difference operator with respect to x Element-wise product of two matrices⊗ Kronecker product of two matrices⊕ Kronecker sum of two matricesσ Spectrum ofFD Finite DifferenceFDM Finite Difference MethodADI Alternating direction implicit1D One-dimensional2D Two-dimensional3D Three-dimensionalCFL condition/number Courant–Friedrichs–Lewy condition/numberPML Perfectly Matched LayerRK4 4th-order Runge-Kutta methodRE Richardson Extrapolation method
ix
Chapter 1
Introduction
Wave equation is one of the most important partial differential equations and it has a lot
of applications in both Math and Physics. For example, the acoustic wave equation with
a non-zero point source function has been widely used to model wave propagation in Geo-
physics. These acoustic wave equations arise in various applications including underground
imaging, seabed exploration, etc. General wave equations also play key roles in Physical
and Engineering models like medical imaging, electromagnetic scattering, non-destructive
testing, meteorology, elastic wave simulation in membranes, waveguides, etc. It is known
that most partial differential equations do not have analytic solutions, thus these equations
are usually solved numerically.
Finite Difference Method (FDM) is one of the most popular numerical methods to solve
partial differential equations numerically due to several reasons such as simple implementa-
tion, high efficiency, etc. The basic idea of finite difference method is to use finite difference
function to approximate derivatives of function. The efficiency and accuracy of finite dif-
ference method are critical, especially when the problem is in large size. In particular,
high-order 1 finite difference methods have attracted increasing attentions recently from re-
searchers. Due to the increasing demand for highly accurate numerical solutions from the
geophysics community, many finite difference methods have been developed to solve acoustic
1In this study, high-order means FDM with spatial accuracy order at least four.
1
wave equations [1, 17, 24, 25, 29, 40]. In addition to the high accuracy, there are many other
benefits from using high-order numerical method, such as the effectiveness in minimizing
dispersion errors [8], fewer grid points being required than the conventional finite difference
methods [7], etc. Furthermore, it has been pointed out that high-order finite difference
methods allow a more coarse sampling rate [13]. A lot of efficient high-order methods have
been developed and implemented with great success [5, 6, 18, 20, 23, 27, 26], to name a few.
However, many high-order FDM are not compact, which leads to difficulty in dealing
with boundary conditions. For example, a typical 4th-order FDM requires a 5-point stencil
to approximate the Laplacian ∆u in 1D cases, a 9-point stencil for 2D cases and a 13-point
stencil for 3D cases, respectively. Therefore, it is hard to implement if there is only one layer
of boundary conditions. To overcome this issue, compact FDM has been developed to reduce
the points needed to approximate derivatives but still keep the high-order accuracy. In [44],
the authors developed a family of fourth-order three-point combined difference schemes to
approximate the first- and second-order spatial derivatives. In [53], the authors introduced
a family of three-level implicit FD schemes which incorporate the locally one-dimensional
method. Some other reported works on compact high-order FD methods for solving acoustic
wave equation can be found in [2, 10, 15, 21] and references therein. However, many existing
compact high-order FD methods were developed based on constant velocity model, thus,
they fail in variable acoustic velocity cases, i.e. acoustic equations with variable coefficients,
not to mention cases with heterogeneous density media.
This study aims to investigate compact high-order finite difference schemes which still
work in variable acoustic velocity cases.
In Chapter 2, an implicit finite difference scheme based on Pade approximation is pro-
posed. This scheme is compact and of 4th-order in both time and space. For three-
dimensional problems, an implicit scheme results in a block tridiagonal system which needs
to be solved at each time step. Direct method for such large block linear system is very ineffi-
cient, therefore, some operator splitting techniques are used to convert the three-dimensional
2
problem into a sequence of one-dimensional problems. This part uses the Alternating Di-
rection Implicit (ADI) method. The development of ADI and related works can be found in
[49, 55, 58, 59, 65]. The conditional stability of this scheme is obtained by an energy method
and together with some functional calculus techniques to estimate the eigenvalues.
In Chapter 3, a different compact scheme with 4th-order spatial accuracy is investigated.
This scheme is based on so-called combined finite difference method proposed by [12, 44]
which also needs the information of derivatives on the boundary for Dirichlet problems. The
boundary conditions needed in the new scheme are obtained from the acoustic wave equation
itself exactly. The stability condition is again obtained by the energy method and the analysis
of eigenvalues of finite difference matrices. This scheme is explicit and of 2nd-order in time,
but the temporal accuracy can be easily improved to 4th-order by Runge-Kutta method or
Richardson Extrapolation method. Comparing to the one in Chapter 2, this scheme avoids
the complicated algebraic manipulation and is much simpler to implement at each time step.
In Chapter 4, the combined finite difference scheme in Chapter 3 is generalized and
applied to acoustic wave equations in media with variable density, i.e. when the normal
Laplacian ∆u is replaced by the divergence form of it given by ∇ ·(
1ρ∇u)
with ρ being the
density function of the media. Most of real-world applications have to deal with variable
density media models. In this scheme, more information of the derivatives of the solution
on the boundary is necessary. The energy method for stability analysis in Chapter 2 and
Chapter 3 is no longer applicable because the discretization of the operator ∇·(
1ρ∇u)
is no
longer a symmetric matrix which is a key assumption in the analysis of the energy method
used in Chapter 2 and Chapter 3. Thus an analogy of von Neumann analysis is used to obtain
the stability condition. The numerical examples in this section also show the effectiveness
of this scheme in solving acoustic wave equations in variable density media with Perfect
Matched Layer (PML) boundary condition.
The last chapter concludes this study.
These new schemes are expected to have wide applications in Geophysics modeling and
3
problems. For example, Full Waveform Inversion (FWI) is a non-linear optimization method
of data-fitting and is one of the most popular methods to solve wave propagation inverse
problems. In seismic modeling, researchers and engineers use full waveform inversion to
obtain information of subsurface properties from seismic data. Full waveform inversion
requires many iterations until convergence during its optimization procedure and solving
wave equations play a key role in each iteration. Thus any improvement of numerical scheme
in solving wave equations will make a big difference for the whole computational framework.
Given the fact that full waveform inversion for seismic modeling usually considers problems
in very large scale, the high accuracy and compactness of the new schemes in this study
can save a lot of computational cost and efforts to collect seismic data, and the stability
guarantee that at each iteration no unexpected numerical error from solving wave equations
will be accumulated into the total error.
Apart from finite difference method, some other numerical methods are also applied to
solve wave equations. For example, Galerkin based schemes, like finite element method,
are also very popular due to their accuracy and flexibility in complicated domain. Spectral
method is another Galerkin based scheme which has a very impressive exponential accuracy,
but it is usually more costly and suffers from serious losses of accuracy on irregular domains.
One of the common trade-off of Galerkin based methods is that they are no longer simple
to implement.
4
Chapter 2
A Compact High Order Alternating
Direction Implicit Method for
Three-dimensional Acoustic Wave
Equations with Variable Coefficients
The main content of this part is taken from [38].
2.1 Acoustic Wave Equation and Pade Approximation
Consider the 3D acoustic wave equation
utt = ν2(x, y, z)(uxx + uyy + uzz) + s(x, y, z, t), (x, y, z, t) ∈ Ω × [0, T ], (2.1)
u(x, y, z, 0) = f1(x, y, z), (x, y, z) ∈ Ω, (2.2)
ut(x, y, z, 0) = f2(x, y, z), (x, y, z) ∈ Ω (2.3)
u(x, y, z, t) = g(x, y, z, t), (x, y, z, t) ∈ ∂Ω × [0, T ], (2.4)
5
where ν(x, y, z) represents the wave velocity. Here Ω ⊂ R3 is a finite computational domain
and s(x, y, z, t) is the source function. We denote c(x, y, z) = ν2(x, y, z) for the sake of simple
notation.
First assume that Ω is a 3D rectangular box : [x0, x1] × [y0, y1] × [z0, z1], which is
discretized into an Nx × Ny × Nz grid with spatial grid sizes hx, hy and hz. Let τ be the
time step size and uni,j,k denote the numerical solution at the grid point (xi, yj, zk) and time
level nτ . Define the standard central difference operators
δ2t u
ni,j,k = un−1
i,j,k − 2uni,j,k + un+1i,j,k ,
δ2xu
ni,j,k = uni−1,j,k − 2uni,j,k + uni+1,j,k,
δ2yu
ni,j,k = uni,j−1,k − 2uni,j,k + uni,j+1,k,
δ2zu
ni,j,k = uni,j,k−1 − 2uni,j,k + uni,j,k+1.
The standard second-order central FD schemes are then given by approximating the second
derivatives in Eq.(2.1). For example,
utt(xi, yj, zk, tn) =1
τ 2δ2t u
ni,j,k +O(τ 2), (2.5)
uxx(xi, yj, zk, tn) =1
h2x
δ2xu
ni,j,k +O(h2
x). (2.6)
The approximations of uyy and uzz are similar to Eq. (2.6).
To improve the method to fourth-order in space, the conventional high-order FD method
was derived by approximating the derivatives using more than three points in one direction,
which results in larger stencil. For instance, if 2M + 1 points are used to approximate uxx,
one can obtain the following formula
uxx(xi, yj, zk, tn) ≈
[a0u
ni,j,k +
M∑m=1
am(uni−m,j,k + uni+m,j,k)
]/h2
x, (2.7)
6
which can be as accurate as (2M)th-order in x. The conventional high-order FD method is
accurate in space but suffers severe numerical dispersion. Another issue is that it requires
more computer memory due to the larger stencil for implicit method. Moreover, more points
are needed to approximate the boundary condition.
To improve the accuracy in time, a class of time-domain high-order FD methods have
been derived by Liu and Sen [63]. The idea of the time-domain high-order FD method is to
determine coefficients using time-space domain dispersion. As a result, the coefficient will be
a function of vτh
. It was noted that in 1D case, the time-domain high-order FD method can
be as accurate as (2M)th-order in both time and space, provided some conditions are satis-
fied, while for multidimensional case, (2M)th-order is also possible along some propagation
directions.
To develop a high-order compact ADI FD scheme, we apply the Pade approximation to
the second-order central FD operators, so the second derivatives utt, uxx, uyy and uzz can
approximated with fourth-order accuracy. For a function f(x) the conventional 2nd-order
approximation for the second derivative is
f(2)i =
1
h2δ2xfi −
h2
12f
(4)i +O(h4) (2.8)
then one also has
f(4)i =
1
h2δ2xf
(2)i −
h2
12f
(6)i +O(h4). (2.9)
Combine the above two approximations one has
f(2)i =
1
h2δ2x(1−
δ2x
12)fi +O(h4). (2.10)
Then together with the approximation
1
1− v= 1 + v +O(v2) (2.11)
7
one can obtain the following Pade approximation based 4th-order approximation for second
derivatives
δ2t
τ 2(1 + 1
12δ2t
)u(xi, yj, zk, tn) = utt(xi, yj, zk, tn) + O(τ 4), (2.12)
δ2x
h2x
(1 + 1
12δ2x
)u(xi, yj, zk, tn) = uxx(xi, yj, zk, tn) + O(h4x). (2.13)
The fourth-order approximations of uyy and uzz can be obtained similarly.
Remark 2.1. Here the operatorδ2t
1+ 112δ2t
is understood as(1 + 1
12δ2t
)−1 · δ2t . It will be seen later
on that(1 + 1
12δ2t
)−1 · δ2t = δ2
t ·(1 + 1
12δ2t
)−1. The similar operators for spatial derivatives
also have this property.
Let λx = τ2
h2x, λy = τ2
h2y, λz = τ2
h2z. Substituting the fourth-order Pade approximations into
Eq. (2.1) gives
δ2t
1 + 112δ2t
uni,j,k =
[λxci,j,k
δ2x
1 + 112δ2x
+ λyci,j,kδ2y
1 + 112δ2y
+ λzci,j,kδ2z
1 + 112δ2z
]uni,j,k+τ
2sni,j,k. (2.14)
Truncation error analysis shows that the algorithm is fourth-order accurate in time and space
with the truncation error O(τ 4 +h4x+h4
y+h4z), provided the solution u(x, y, z, t) and c(x, y, z)
satisfy certain smoothness conditions. As shown in [61], the difficulty to develop high-order
compact scheme for wave equation with non-constant velocity is that, one cannot simply
multiply the operator
(1 +
δ2t
12
)(1 +
δ2x
12
)(1 +
δ2y
12
)(1 +
δ2z
12
)(2.15)
to both sides of Eq. (2.14) to cancel the fractional operators (1 +δ2t12
)−1
, (1 + δ2x12
)−1
, (1 +δ2y12
)−1
and (1 + δ2z12
)−1
because that will be difficult in implementation due to ci,j,k being non-
constant. To overcome this difficulty, we develop an algebraic strategy which will be de-
scribed in the next section.
8
2.2 Derivation of the Compact High-order ADI Method
Now we extend the novel algebraic manipulation introduced in [61] to the three-dimensional
acoustic wave equation with variable velocity. We first demonstrate the difficulty in apply-
ing the Pade approximation to the finite difference operator for solving the acoustic wave
equation with non-constant velocity. Applying the operator in Eq. (2.15) to Eq. (2.14)
yields
(1 +
1
12δ2x
)(1 +
1
12δ2y
)(1 +
1
12δ2z
)δ2t u
ni,j,k
= λx
(1 +
1
12δ2t
)(1 +
1
12δ2x
)(1 +
1
12δ2y
)(1 +
1
12δ2z
)ci,j,k
δ2x
1 + 112δ2x
uni,j,k
+λy
(1 +
1
12δ2t
)(1 +
1
12δ2x
)(1 +
1
12δ2y
)(1 +
1
12δ2z
)ci,j,k
δ2y
1 + 112δ2y
uni,j,k
+λz
(1 +
1
12δ2t
)(1 +
1
12δ2x
)(1 +
1
12δ2y
)(1 +
1
12δ2z
)ci,j,k
δ2z
1 + 112δ2z
uni,j,k
+τ 2
(1 +
1
12δ2t
)(1 +
1
12δ2x
)(1 +
1
12δ2y
)(1 +
1
12δ2z
)sni,j,k. (2.16)
We use the first term on the right-hand side to illustrate the issue here. Since (1 + δ2x12
),
(1 +δ2y12
) and (1 + δ2z12
) are commutative, we change the order of the three finite difference
operators to obtain
λx
(1 +
1
12δ2t
)(1 +
1
12δ2x
)(1 +
1
12δ2y
)(1 +
1
12δ2z
)ci,j,k
δ2x
1 + 112δ2x
uni,j,k =
λx
(1 +
1
12δ2t
)(1 +
1
12δ2y
)(1 +
1
12δ2z
)(1 +
1
12δ2x
)ci,j,k
δ2x
1 + 112δ2x
uni,j,k. (2.17)
As discussed previously, when c(x, y, z) is non-constant in terms of x, the operator (1 +
112δ2x) and ci,j,k are not commutative. Hence,
(1 + 1
12δ2x
)ci,j,k 6= ci,j,k
(1 + 1
12δ2x
). Therefore,
9
the operator(1 + 1
12δ2x
)does not cancel the operator
(1 + 1
12δ2x
)−1. In other words,
λx
(1 +
1
12δ2t
)(1 +
1
12δ2y
)(1 +
1
12δ2z
)(1 +
1
12δ2x
)ci,j,k
δ2x
1 + 112δ2x
uni,j,k
6= λx
(1 +
1
12δ2t
)(1 +
1
12δ2y
)(1 +
1
12δ2z
)ci,j,k δ
2xu
ni,j,k. (2.18)
To solve this problem, a novel factorization technique is used to preserve the compactness
and fourth-order convergence of the numerical scheme. Applying (1+δ2t12
) to Eq. (2.14) yields
δ2t u
ni,j,k = ci,j,k
[λx
(1 +
δ2t
12
)δ2x
1 + δ2x12
+ λy
(1 +
δ2t
12
)δ2y
1 +δ2y12
+
λz
(1 +
δ2t
12
)δ2z
1 + δ2z12
]uni,j,k + τ 2
(1 +
δ2t
12
)sni,j,k. (2.19)
Collecting the term δ2t u
ni,j,k, we have
[1− λxci,j,k
12
δ2x
1 + 112δ2x
− λyci,j,k12
δ2y
1 + 112δ2y
− λyci,j,k12
δ2z
1 + 112δ2z
]δ2t u
ni,j,k
= ci,j,k
[λxδ
2x
1 + δ2x12
+λyδ
2y
1 +δ2y12
+λzδ
2z
1 + δ2z12
]uni,j,k + τ 2
(1 +
δ2t
12
)sni,j,k. (2.20)
Factoring the left-hand side of Eq. (2.20) yields
[1− ci,j,k
12
λxδ2x
1 + 112δ2x
]·
[1− ci,j,k
12
λyδ2y
1 + 112δ2y
]·[1− ci,j,k
12
λzδ2z
1 + 112δ2z
]δ2t u
ni,j,k =
ci,j,k
[λxδ
2x
1 + 112δ2x
+λyδ
2y
1 + 112δ2y
+λzδ
2z
1 + 112δ2z
]uni,j,k + τ 2
(1 +
δ2t
12
)sni,j,k + ERR, (2.21)
10
where the factorization error is given by
ERR =λxλy144
ci,j,kδ2x
1 + δ2x12
ci,j,kδ2y
1 +δ2y12
δ2t u
ni,j,k +
λyλz144
ci,j,k δ2y
1 +δ2y12
ci,j,k δ2z
1 + δ2z12
δ2t u
ni,j,k +
λxλz144
ci,j,k δ2x
1 + δ2x12
ci,j,k δ2z
1 + δ2z12
δ2t u
ni,j,k
−λxλyλz1728
ci,j,k δ2x
1 + δ2x12
ci,j,k δ2y
1 +δ2y12
ci,j,k δ2z
1 + δ2z12
δ2t u
ni,j,k. (2.22)
It is simply the trick that
(1− a)(1− b)(1− c) = (1− a− b− c+ ab+ bc+ ac− abc). (2.23)
Using Taylor series, one can verify that ERR = O(τ 6), provided that c(x, y, z) and u(x, y, z, t)
satisfy some conditions on smoothness. The result regarding the order of the truncation error
is included in the following theorem.
Theorem 2.2. Assume that u(x, y, z, t) ∈ C6,6,6,6x,y,z,t (Ω× [0, T ]) is the solution of the acoustic
wave equation defined by Eqs. ( 2.1 - 2.4), and the coefficient satisfies the smoothness
condition c(x, y, z) ∈ C2,2,2x,y,z(Ω). Then the truncation error given in Eq. (2.22) satisfies
ERR = O(τ 6) +O(h6x) +O(h6
y) +O(h6z),
where τ, hx, hy and hz are the step sizes in time, x, y and z, respectively.
Proof. A detailed proof of the theorem is given in Appendix A
11
Remark 2.3. If Eq. (2.20) is factorized in a different order of δ2x, δ
2y and δ2
z , for instance as
[1− ci,j,k
12
λyδ2y
1 + 112δ2y
]·[1− ci,j,k
12
λxδ2x
1 + 112δ2x
]·[1− ci,j,k
12
λzδ2z
1 + 112δ2z
]δ2t u
ni,j,k =
ci,j,k
[λxδ
2x
1 + 112δ2x
+λyδ
2y
1 + 112δ2y
+λzδ
2z
1 + 112δ2z
]uni,j,k + τ 2
(1 +
δ2t
12
)sni,j,k + ERR, (2.24)
then the factoring error ERR is given by
ERR =λy144
ci,j,kδ2y
1 + 112δ2y
λx ci,j,kδ2x
1 + 112δ2x
δ2t u
ni,j,k
+λy144
ci,j,kδ2y
1 + 112δ2y
λx ci,j,kδ2x
1 + 112δ2x
δ2t u
ni,j,k
+λy144
ci,j,kδ2y
1 + 112δ2y
λx ci,j,kδ2x
1 + 112δ2x
δ2t u
ni,j,k
− 1
1728λy ci,j,k
δ2y
1 + 112δ2y
λx ci,j,kδ2x
1 + 112δ2x
δ2t u
ni,j,k, (2.25)
which has the same error estimation as that defined in Eq. (A.16).
Ignoring the factoring error ERR in Eq. (2.21) leads to the following compact fourth-
order FD method
[1− λxci,j,k
12
δ2x
1 + δ2x12
]·
[1− λyci,j,k
12
δ2y
1 +δ2y12
]·
[1− λzci,j,k
12
δ2z
1 + δ2z12
]δ2t u
ni,j,k
= ci,j,k
[λxδ
2x
1 + δ2x12
+λyδ
2y
1 +δ2y12
+λzδ
2z
1 + δ2z12
]uni,j,k + τ 2
(1 +
δ2t
12
)sni,j,k. (2.26)
12
With ADI method, Eq. (2.26) can be efficiently solved in three steps
(1− λxci,j,k
12
δ2x
1 + δ2x12
)u∗∗i,j,k =
[ci,j,k λxδ
2x
1 + δ2x12
+ci,j,k λyδ
2y
1 +δ2y12
+ci,j,k λzδ
2z
1 + δ2z12
]uni,j,k
+τ 2
(1 +
δ2t
12
)sni,j,k, 2 ≤ j ≤ Ny − 1, 2 ≤ k ≤ Nz − 1, (2.27)(
1− λyci,j,k12
δ2y
1 +δ2y12
)u∗i,j,k = u∗∗i,j,k, 2 ≤ i ≤ Nx − 1, 2 ≤ k ≤ Nz − 1, (2.28)(
1− λzci,j,k12
δ2z
1 + δ2z12
)δ2t u
ni,j,k = u∗i,j,k, 2 ≤ i ≤ Nx − 1, 2 ≤ j ≤ Ny − 1. (2.29)
Apparently the three equations are difficult to implement due to the three operators(1 + δ2x
12
)−1
,(
1 +δ2y12
)−1
and(
1 + δ2z12
)−1
. To overcome this problem, we apply the following
strategy. Firstly, divide both sides of Eq. (2.27) by ci,j,k, then multiply(
1 + δ2x12
), we have
[(1 +
δ2x
12
)1
ci,j,k− λx
12δ2x
]u∗∗i,j,k = τ 2
(1 +
δ2x
12
)(1 +
δ2t
12
)sni,j,kci,j,k
+
[λxδ
2x + λy
(1 +
δ2x
12
)δ2y
1 +δ2y12
+ λz
(1 +
δ2x
12
)δ2z
1 + δ2z12
]uni,j,k. (2.30)
Eq. (2.30) is still hard to implement because of the termsδ2y
1+δ2y12
and δ2z
1+δ2z12
. Substituting
δ2y
1+δ2y12
uni,j,k with δ2y
(1− δ2y
12
)uni,j,k,
δ2z
1+δ2z12
uni,j,k with δ2z
(1− δ2z
12
)uni,j,k, respectively, we obtain
[(1 +
δ2x
12
)1
ci,j,k− λx
δ2x
12
]u∗∗i,j,k = τ 2
(1 +
δ2x
12
)(1 +
δ2t
12
)sni,j,kci,j,k
+[λxδ
2x + λy
(1 +
δ2x
12
)δ2y
(1−
δ2y
12
)+ λz
(1 +
δ2x
12
)δ2z
(1− δ2
z
12
)]uni,j,k,
for 2 ≤ j ≤ Ny − 1, 2 ≤ k ≤ Nz − 1. (2.31)
Note the difference between Eq. (2.30) and Eq. (2.31) is O(h6y + h6
z), therefore, the method
is still fourth-order in space. It is worth to mention that the right-hand side of Eq. (2.31)
includes larger stencil in both y and z directions. Furthermore, larger stencil needs values of
13
uni,j,k outside the boundary when j = 2, Ny−1 and k = 2, Nz−1. To overcome this problem,
we use one-sided approximation to approximate the values outside of the boundary. For
example, uni,0,k is approximated by a linear combination of uni,1,k, · · · , uni,4,k with fourth-order
accuracy. This boundary treatment is not complicate in terms of implementation, since
it only involves the values at time level n, which is known. Similarly, dividing ci,j,k then
multiplying(
1 +δ2y12
)to both sides of Eq. (2.28) lead to
[(1 +
δ2y
12
)1
ci,j,k− λy
δ2y
12
]u∗i,j,k =
(1 +
δ2y
12
)u∗∗i,j,kci,j,k
(2.32)
for i = 2, 3, · · · , Nx − 1, k = 2, 3, · · · , Nz − 1.
Finally, Eq. (2.29) can be transformed to the equivalent linear system
[(1 +
δ2z
12
)1
ci,j,k− λz
δ2z
12
]δ2t u
ni,j,k =
(1 +
δ2z
12
)u∗i,j,kci,j,k
, (2.33)
for i = 2, 3, · · · , Nx − 1, j = 2, 3, · · · , Ny − 1.
It is noted that Eq. (2.33) is equivalent to a three-level FD scheme
[(1 +
δ2z
12
)1
ci,j,k− λz
12δ2z
]un+1i,j,k =
[(1 +
δ2z
12
)1
ci,j,k− λz
12δ2z
](2uni,j,k − un−1
i,j,k)
+
(1 +
δ2z
12
)u∗i,j,kci,j,k
, i = 2, 3, · · · , Nx − 1, j = 2, 3, · · · , Ny − 1. (2.34)
By now the three linear systems defined in Eqs. (2.31, 2.32, 2.34) can be efficiently solved
using tridiagonal matrix algorithm. Here some one-sided fourth-order approximations are
needed for boundary condition approximations in these equation systems. For example, in
Eq. (2.33), the following fourth-order one-sided approximations are used to approximate
14
u∗i,j,1 and u∗i,j,Nz , respectively:
u∗i,j,1 = 4u∗i,j,2 − 6u∗i,j,3 + 4u∗i,j,44− u∗i,j,5,
u∗i,j,Ny = 4u∗i,j,Nz−1 − 6u∗i,j,Nz−2 + 4u∗i,j,Nz−3 − u∗i,j,Nz−4,
for i = 2, 3, · · · , Nx − 1, j = 2, 3, · · · , Ny − 1.
The boundary conditions for Eq. (2.32) can be obtained by setting j = 1 and j = Ny in
Eq. (2.33), respectively.
(1 +
δ2z
12
)u∗i,1,kci,1,k
=
[(1 +
δ2z
12
)1
ci,1,k− λz
12δ2z
]δ2t u
ni,1,k, (2.35)(
1 +δ2z
12
)u∗i,Ny ,k
ci,Ny ,k=
[(1 +
δ2z
12
)1
ci,Ny ,k− λz
12δ2z
]δ2t u
ni,Ny ,k. (2.36)
Solving the two tridiagonal linear systems we can get the boundary conditions for Eq. (2.32).
Similarly, the boundary conditions needed by Eq. (2.31) can be obtained by letting i = 1
and i = Nx, respectively.
The new compact ADI method defined in Eq. (2.34) is a three-level FD scheme. There-
fore, two initial conditions are needed at t = 0 and t = τ . To approximate the initial
condition at t = τ with fourth-order accuracy, we expand u(xi, yj, zk, t) by the Taylor series
at t = 0 and obtain the following fourth-order approximation
u1i,j,k = u0
i,j,k + τ∂u
∂t|0i,j,k +
τ 2
2
∂2u
∂t2|0i,j,k +
τ 3
6
∂3u
∂t3|0i,j,k +
τ 4
24
∂4u
∂t4|0i,j,k +O(τ 5), (2.37)
where the high-order derivatives are derived using the method in [61].
Now we state and prove the main result on the convergence of the compact ADI FD
scheme defined in Eq. (2.26).
Theorem 2.4. Assume that u(x, y, z, t) ∈ C6,6,6,6x,y,z,t (Ω× [0, T ]) is the solution of the acoustic
wave equation defined in Eqs. ( 2.1 - 2.4), and the coefficient function satisfies the smooth
15
condition c(x, y, z) ∈ C2,2,2x,y,z(Ω). Then the compact ADI FD scheme defined in Eq. (2.26) is
fourth-order accurate in time and space with the truncation error O(τ 4 + h4x + h4
y + h4z).
Proof. According to Theorem 3.1, if u(x, y, , z, t) and c(x, y, z) are sufficiently smooth, the
difference between the numerical scheme defined in Eq. (2.26) and the numerical scheme
defined in Eq. (2.20) is O(τ 6) +O(h6x) +O(h6
y) +O(h6z).
On the other hand, it is known[60] that the compact Pade approximation FD method
defined in Eq. (2.26) is fourth-order in time and space, with the truncation error O(τ 4) +
O(h4x) +O(h4
y) +O(h4z).
Moreover, one can see that the truncation errors caused by the substitutions
δ2x
(1 + δ2x12
)→ δ2
x
(1− δ2
x
12
),
δ2y
(1 +δ2y12
)→ δ2
y
(1−
δ2y
12
),
δ2z
(1 + δ2z12
)→ δ2
z
(1− δ2
z
12
)
in Eq. (2.31) are O(h6x), O(h6
y) and O(h6z), respectively. Base on these, the new compact
ADI FD scheme is fourth-order in time and space.
2.3 Stability Analysis
It is important that a numerical method is stable when it is applied to solve time-dependent
problem. Most of the FD schemes for solving the acoustic wave equation are conditionally
stable and subject to constraints on time step. The popular Von Neumann analysis is
applicable for constant velocity case only, therefore, in this paper we adopted the energy
method in [2] to analyze and prove the stability of the new method.
For the sake of simplicity, assume zero source for the wave equation. Consider the Pade
approximation based fourth-order finite difference scheme
δ2t
1 + 112δ2t
uni,j,k = ci,j,k
[λxδ
2x
1 + 112δ2x
+λyδ
2y
1 + 112δ2y
+λzδ
2z
1 + 112δ2z
]uni,j,k, (2.38)
16
where λx = τ2
h2x, λy = τ2
h2y, λz = τ2
h2z. For simplicity we assume h = hx = hy = hz, λ = λx =
λy = λz, ω = λci,j,k =τ2ci,j,kh2
. Note that ω is a grid function but independent of time. Then
the above scheme becomes
1
ω
δ2t
1 + 112δ2t
uni,j,k =
[δ2x
1 + 112δ2x
+δ2y
1 + 112δ2y
+δ2z
1 + 112δ2z
]uni,j,k. (2.39)
If we let
L =δ2x
1 + 112δ2x
+δ2y
1 + 112δ2y
+δ2z
1 + 112δ2z
= Tx + Ty + Tz,
The scheme can be written as
1
ω
δ2t
1 + 112δ2t
un = L un, (2.40)
where un is the numerical solution at time level tn:
un = (uni,j,k)N×N×N .
Here we assume that Nx = Ny = Nz = N .
To prove the stability, we first state the following lemma, which can be found in standard
functional analysis textbook such as [70].
Lemma 2.5. If f is continuous function and A is a self-adjoint operator, then
σ(f(A)) = f(σ(A)).
Lemma 2.6. Tx, Ty and Tz are self-adjoint operators, so is the sum L = Tx + Ty + Tz.
Proof. Let
f(s) =s
1 + 112s,
17
then Tx = f(δ2x). Note that we can write
f(s) = s ·(
1 +1
12s
)−1
=
(1 +
1
12s
)−1
· s, (2.41)
which means
δ2x ·(
1 +1
12δ2x
)−1
= f(δ2x) =
(1 +
1
12δ2x
)−1
· δ2x. (2.42)
Thus, if we can prove that both δ2x and (1 + 1
12δ2x)−1 are self-adjoint, then Tx = f(δx), as a
product of two commutative self-adjoint operators, is also self-adjoint. It is clear that both
δ2x and 1 + 1
12δ2x are self-adjoint, then (1 + 1
12δ2x)−1, as an inverse of a self-adjoint operator, is
self-adjoint. The proofs for Ty and Tz are similar. Finally L is self-adjoint since it is a sum
of three self-adjoint operators.
The spectrum of δ2x with the homogeneous Dirichlet boundary condition is given by
σ(δ2x) =
−4 sin2
(j π
2(N + 1)
)⊂ (−4, a(h)],
where N is the number of grid points in the x-direction, j = 1, · · · , N , a(h) = −4 sin2(πh2
).
Then lemma 2.5 implies that the spectrum of Tx is given by
σ(Tx) = f(σ(δ2x)) =
f
(−4 sin2
(j π
2(N + 1)
)),
where f(s) = s1+ 1
12s
and j = 1, · · · , N . Since f(s) is increasing when s > −12, we have
σ(Tx) ⊂ (−6, f(a(h))].
We have f(a(h)) < 0 since −12 < a(h) < 0 when h is small enough. Note that the operators
Tx, Ty and Tz are actually hermitian matrices, and the fact that the operator L corresponds
18
to a homogeneous Dirichlet problem, then (for instance, see [54])
σ(L) ⊂ (−18, 3f(a(h))].
Thus, we obtained the coercive condition, which is a direct result of its spectrum estimate,
m = −3f(a(h)) ≤ −L ≤ 18 = M. (2.43)
Remark 2.7. Note that h is the grid size and should be small enough, then
a(h) = −4 sin2(πh
2
)≈ −π2h2
Now we state the main result on the stability of the new method in the following theorem.
Theorem 2.8. Assume that the solution of the acoustic wave equation Eq. (2.1) is suffi-
ciently smooth, the new scheme is stable if
max1≤i,j,k≤N
νi,j,k τ
h<
1√3.
Proof. Here we follow the strategy of [2] to prove the stability. Firstly denote the l2 norm
by ‖ · ‖, the inner product on l2 by 〈·, ·〉. Recall that un is the numerical solution at tn.
From (2.40), since δ2t commutes with L, we have
1
ωδ2t u
n − 1
12Lδ2
t un = Lun. (2.44)
Define vn = un−un−1, then δ2t u
n = vn+1−vn, vn+1 +vn = un+1−un−1. Taking inner product
with vn+1 + vn on both sides of Eq. (2.44), noting that L is self-adjoint, we have
〈 1ω
(vn+1 − vn), vn+1 + vn〉 − 1
12〈L(vn+1 − vn), vn+1 + vn〉
=〈Lun, un+1 − un−1〉(2.45)
19
Expanding the right-hand side of Eq. (2.45) gives
〈Lun, un+1 − un−1〉
=1
4
[〈Lvn, vn〉 − 〈Lvn+1, vn+1〉 − 〈L(un + un−1), un + un−1〉
+〈L(un+1 + un), un+1 + un〉] (2.46)
Combining (2.45) and (2.46), noting that the cross terms in (2.45) are eliminated since L is
self-adjoint, we have
1
ω〈vn, vn〉+
1
6〈Lvn, vn〉 − 1
4〈L(un + un−1), un + un−1〉
=1
ω〈vn+1, vn+1〉+
1
6〈Lvn+1, vn+1〉 − 1
4〈L(un+1 + un), un+1 + un〉
(2.47)
If we define
Sn =1
ω‖vn‖2 +
1
6〈Lvn, vn〉 − 1
4〈L(un + un−1), un + un−1〉,
then the above equality is exactly
Sn = Sn+1.
By the coercivity of −L, m ≤ −L ≤M , we have
Sn ≥1
ω‖vn‖2 − M
6‖vn‖2 +
m
4‖un + un−1‖2 (2.48)
and
Sn ≤1
ω‖vn‖2 − m
6‖vn‖2 +
M
4‖un + un−1‖2. (2.49)
20
Thus, Sn is equivalent to the energy given by
‖vn‖2 + ‖un + un−1‖2
=‖un − un−1‖2 + ‖un + un−1‖2
=2‖un‖2 + 2‖un−1‖2
if and only if
1
ω>M
6,
i.e.
maxi,j,k
ν2i,j,k ·
τ 2
h2<
6
M=
1
3⇒ max
i,j,kνi,j,k ·
τ
h<
1√3. (2.50)
since M = 18 by Eq. (2.43).
Denoted by en, the error at time tn, the above stability analysis shows that the energy
of the error ‖en‖2 + ‖en−1‖2 conserves during the solving process, which means the scheme
is conditionally stable, as long as the stability condition in Eq. (2.50) is satisfied.
For comparison, according to [62], the stability condition for 3-D problem using the
standard second-order difference scheme is
maxi,j,k
νi,j,k ·τ
h<
1
3.
Moreover, the stability condition for the conventional fourth-order FD scheme(it is second-
order in time) is
maxi,j,k
νi,j,k ·τ
h<
1
2. (2.51)
Apparently, the new method has better stability with a larger CFL number. Although
in each time step a sequence of tridiagonal linear systems need to be solved to march the
numerical solution, the high-order ADI method outperforms other existing methods in terms
21
of the overall efficiency. One can also see that estimated upper bound of τh
is sharp, as
demonstrated in the second numerical example.
2.4 Numerical Examples
In this section three numerical examples are solved by the new method to demonstrate the
efficiency and accuracy. The exact solutions of the first and second examples are available,
so the numerical errors can be calculated to validate the order of convergence and stability
condition of the new method. In the third example, the acoustic wave equation with the
Ricker’s wavelet source is solved to demonstrate that the new method is effective in sup-
pressing numerical dispersion and efficient and accurate in simulating wave propagation in
heterogeneous media. It is worthwhile to mention that in the following numerical examples,
all numerical errors are calculated using maximal norm, although the stability and error
analysis were conducted in L2 norm. Since the maximal and L2 norms are equivalent, all
conclusions confirmed in maximal norm hold in L2 norm.
2.4.1 Example 1
In this example, the acoustic wave equation is defined on a rectangular domain Ω = [0, π]×
[0, π]× [0, π], and t ∈ [0, T ],
utt =
[1 +
(xπ
)2
+(yπ
)2
+( zπ
)2]
∆u+ s(x, y, z, t), (x, y, z, t) ∈ Ω× [0, T ],
u(x, y, z, 0) = sin(x) sin(y) sin(z), (x, y, z) ∈ Ω,
ut(x, y, z, 0) = 0, (x, y, z) ∈ Ω,
u|∂Ω = 0, (x, y, z, t) ∈ ∂Ω× [0, T ],
22
with the source function given by
s(x, y, z, t) =
[3 + 2
(xπ
)2
+ 2(yπ
)2
+ 2( zπ
)2]
cos(t) sin(x) sin(y) sin(z)
and the analytical solution given by
u(x, y, z, t) = cos(t) sin(x) sin(y) sin(z).
It is noted that this problem has zero boundary condition, which is selected to simplify
programming. A more general example with non-zero boundary condition will be solved in
the next example. In all numerical simulations, the domain Ω is divided into an Nx×Ny×Nz
grid. The time domain is uniformly divided into Nt subdomains. To simplify the discussion,
uniform grid size h is used in x, y and z directions. To validate the fourth-order convergence
in space, we fixed τ = 0.0025 so the temporal truncation error is negligible. The errors in
maximal norm obtained by using different h are included in Table 2.1, which clearly show
that the new method is fourth-order accurate in space. Here the numerical order is calculated
using the following formula
Conv. Order =log(E(h1)/E(h2))
log(h1/h2),
where the numerical error E(h) is defined by
E(h) = max1≤i≤Nx1≤j≤Ny1≤k≤Nz
∣∣u(xi, yj, zk, T )− uNti,j,k∣∣ .
Here u(xi, yj, zk, T ) is the exact solution of u at the grid point (xi, yj, zk) and time t = T , and
uNti,j,k is the numerical solution at the same grid point and time t = T that is computed using
stepsize h. The numerical error E(h, τ) included in Tables (2.2 - 2.5) is defined similarly,
with the only difference that the time step size τ is changed when h is changed.
23
It is clear from Table 2.1 that the errors are reduced roughly by a factor 16 when h is
reduced by a factor 2. We notice that the convergence order is slightly lower than fourth-
order, due to the round-off errors. To show that the method is fourth-order accurate in time,
Table 2.1: Numerical errors in maximal norm for Example 2.4.1 with τ = 0.0025 at T = 1.h π/10 π/16 π/20 π/32 π/40
E(h) 4.3196e-04 5.4662e-05 2.1191e-05 2.5748e-06 9.9448e-07Conv. Order - 4.3982 4.2466 4.4847 4.2632CPU time(s) 5.4699 9.1399 15.1600 42.3399 68.5999
h and τ are simultaneously reduced by the same factor to ensure that the CFL condition
is satisfied, since using very small h to verify the order in time will violate the stability
condition. Therefore, we verify the order of convergence in time through the following
argument. Suppose the numerical scheme is pth-order accurate in time and fourth-order in
space, with p < 4, halving h and τ several times, the truncation error in time will become
the dominating error, thus the total error will be reduced by a factor of 2p < 16 when h
and τ been halved. In the following numerical test cases, we start from h = π/10, τ = 1/16
(the parameters are chosen to satisfy the stability condition) and each time we halve both
h and τ . The result in Table 2.2 clearly indicates that the total error is reduced by a
factor 16 (roughly) when h and τ are halved, which confirmed that the convergence order
in time is fourth-order. Furthermore, it is interesting that the convergence order is slightly
Table 2.2: Numerical errors in maximal norm for Example 2.4.1 with various h and τ .(h, τ) (π/10, 1/16) (π/20, 1/32) (π/40, 1/64) (π/80, 1/128)E(h, τ) 4.0768e-04 2.0790e-05 9.5059e-07 4.2791e-08E(h,τ)
E(h/2,τ/2)- 19.6094 21.8706 22.2147
Conv. Order - 4.2934 4.45092 4.4734
higher than 4. One possible explanation is that the truncation errors are canceled during
the computation. The new method is an implicit scheme, so the computational cost in each
time step is higher than that of the explicit method, however, the overall computational
24
efficiency has been greatly improved due to the high-order convergence and larger time step
size τ being used.
2.4.2 Example 2
In the second example, we solve a more general acoustic wave equation defined on [0, π]×
[0, π] × [0, π] × [0, T ] with non-zero boundary conditions. The analytical solution for the
following equation
utt =(1 + sin2 x+ sin2 y + sin2 z
)(uxx + uyy + uzz) + s(x, y, z, t)
is given by
u(x, y, z, t) = e−t cos(x) cos(y) cos(z),
where s(x, y, z, t) = (4 + 3(sin2 x+ sin2 y + sin2 z))e−t cos(x) cos(y) cos(z). For this example,
the boundary conditions are non-zero. For example, on the plane x = 0, the boundary
condition is given by
u(0, y, z, t) = e−t cos(y) cos(z), (y, z) ∈ [0, π]× [0, π], t > 0.
To demonstrate the fourth-order convergence in space and time, we reduce τ and h simulta-
neously by the same factor and record the numerical errors in Table 2.3.
Table 2.3: Numerical errors in maximal norm for Example 2.4.2 with various h and τ .(h, τ) (π/16, 1/20) (π/32, 1/40) (π/64, 1/80) (π/128, 1/160)E(h, τ) 5.1391e-05 4.2849e-06 3.9569e-07 2.9088e-08E(h,τ)
E(h/2,τ/2)- 11.9935 10.8289 13.6032
Conv. Order - 3.5842 3.4368 3.7659CPU time(s) 6.83 22.09 128.04 806.29
We then show that the estimated CFL constraint is sharp. According to the proof, the
method is conditionally stable with the CFL condition maxi,j,k νi,j,kτh< 1√
3. The maximum
25
value of ν is given by
νmax =√
max0≤x,y,z≤π
(1 + sin2 x+ sin2 y + sin2 z) = 2.
Therefore, the following stability condition
τ
h<
1
2√
3≈ 0.288675134594813
is required.
First we choose τ and h such that τh
is slightly less than 12√
3so the stability condition
given by Theorem 2.8 is satisfied. For all test cases in Table 2.4, we have
τ
h=
9
10π≈ 0.286478897565412 <
1
2√
3≈ 0.288675134594813.
The numerical results in Table 2.4 confirmed that the numerical method is stable when the
stability condition is met. As can be seen, when τ and h are reduced, the numerical error
is also reduced, showing a convergence order between 3.25 and 4. The noticeable deviation
from a perfect fourth oder in convergence is possibly caused by the fact that τh
is very close to
the CFL condition. Thus, the CFL condition might be slightly violated due to some random
roundoff error, which then deteriorates the convergence order. Nevertheless, the numerical
results in Table 2.4 clearly show that the method is stable when τh< 1
2√
3.
Table 2.4: Numerical errors in maximal norm for Example 2.4.2 with τh< 1
2√
3.
(h, τ) (π/18, 1/20) (π/36, 1/40) (π/54, 1/60) (π/72, 1/80)E(h, τ) 3.3689e-05 2.7867e-06 6.9049e-07 2.7001e-07E(h1,τ1)E(h2,τ2)
- 11.9935 4.0358 2.5573
Conv. Order - 3.5842 3.4410 3.2638
We then numerically validate that the estimated CFL condition is a necessary condition
for stability. To this end, we choose τ and h such that τh
is slightly greater than 12√
3, thus,
26
the stability condition given by Theorem 2.8 is slightly violated. As shown in Table 2.5, the
ratio τh
= 1920π≈ 0.302394 > 1
2√
3. The numerical results clearly show that the method is
not stable, as the numerical solution is not convergent, when τ and h are reduced. Instead,
when τ and h are reduced, the maximal error increases and goes to infinity. Here × in the
Table 2.5: Numerical errors in maximal norm for Example 2.4.2 with τh> 1
2√
3.
(h, τ) (π/19, 1/20) (π/38, 1/40) (π/57, 1/60) (π/76, 1/80)E(h, τ) 8.5884e-05 1.1864e-05 7.2660e-01 1.3165e+06E(h1,τ1)E(h2,τ2)
- 7.2387 × ×Conv. Order - 2.8557 × ×
table means that no convergence is obtained.
2.4.3 Example 3
To demonstrate the efficiency of the new method and show the effectiveness in suppressing
numerical dispersion, we solve a realistic problem in which the seismic wave is generated by
a Ricker wavelet source located at the centre of a three-dimensional domain [0m, 1280m]×
[0m, 1280m]× [0m, 800m]. The velocity model is given by
ν(x, y, z) = 1200 + 400
(x
xmax
)2
+ 100
(y
ymax
)2
+ 800
(z
zmax
)2
,
where xmax = 1280m, ymax = 1280m and zmax = 800m, respectively. Therefore, the maximal
and minimal wave speeds over the domain are
νmax = maxx,y,z
ν(x, y, z) = 2500m/s
and
νmin = minx,y,z
ν(x, y, z) = 1200m/s,
27
respectively. The Ricker wavelet source function is given by
s(x, y, z, t) = δ(x− x0, y − y0, z − z0)[1− 2π2f 2
p (t− dr)2]e−π
2f2p (t−dr)2 ,
where fp = 10Hz is the dominant frequency, dr = 0.5/fp is the temporal delay to ensure zero
initial conditions. The centre of the domain is located at (x0, y0, z0) = (640, 640m, 400m).
The space and time step sizes are chosen to satisfy the CFL condition. Moreover, the Nyquist
sampling theorem states that the sampling frequency should be at least twice the highest
frequency contained in the signal to avoid aliasing. As a rule of thumb, at least 10 grid points
per wavelength are required in finite difference discretization. Simple calculation shows that
the minimal wavelength is given by νmin/fp = 120m, which sets the upper limit for h as
hmax = 12m. On the other hand, the CFL condition indicates that
νmaxτ
h<
1√3⇒ τ
h<
1
2500√
3
For all numerical simulations of this example, the uniform grid h = 10m and τ = 0.001s are
used to ensure stability and avoid aliasing.
We plot the wavefield snapshots for central slices in x−, y− and z− directions at t =
0.2s, t = 0.3s, t = 0.4s and t = 0.5s in Figs. (2.1 - 2.3), respectively. Apparently the
computed wave fronts accurately reflect the wave velocity. For example, in Fig. 2.1 the
simulated wavefields accurately describe the wave propagation. The x-sections are the slices
of the 3D wavefields at x = xmax/2. Since the wave velocity increases more rapidly in z
direction than it does in y direction, we clearly see that the wavefront moves faster in z
direction. Moreover, the wave velocity is a monotone increasing function of y and z, the
wavefronts moves faster at the right side than at the left side, and moves faster in the lower
side than in the upper side of the domain.
Fig. 2.2 shows the snapshots of the wavefields when y = ymax/2 is fixed. Again, the
velocity is a function of two variables x and z, and the velocity increases faster in z. Clearly
28
the wavefronts moves faster in z direction. While in the same direction, the wave propagates
faster when the variable is at the large side. For example, the wavefront hits the right
boundary before it hits the left boundary.
Finally, in Fig. 2.3, the wavefields slices demonstrate similar wave propagations, which
is also expected from theoretical analysis. As shown in these figures, there is no visible
numerical dispersions, which indicated that the new numerical algorithm is accurate and
effective in suppressing numerical dispersion.
Figure 2.1: Snapshots of x-section of wavefields computed by the new 4th-order compactmethod at (a) t = 0.2s; (b) t = 0.3s; (c) t = 0.4s; (d) t = 0.5s.
2.5 Conclusion
A compact fourth-order ADI FD scheme has been developed to solve the three-dimensional
acoustic wave equation in heterogeneous media. The new method is efficiently implemented
using the ADI technique, which splits the original three-dimensional problem into a series
29
Figure 2.2: Snapshots of y-section of wavefields computed by the new 4th-order compactmethod at (a) t = 0.2s; (b) t = 0.3s; (c) t = 0.4s; (d) t = 0.5s.
of one-dimensional problems. The fourth-order convergence in time and space has been
validated by two numerical examples for which the exact solutions are available. A more
realistic problem has been solved to demonstrate that the new method is robust, accurate
and efficient in seismic wave propagation simulation. Moreover, the conditional stability of
the new method has been rigidly proved for the variable coefficient case. It has been shown
that the new method has a larger CFL number than other conventional finite difference
methods. Several numerical tests has been performed to verify that the estimated upper
bounds of CFL is sharp. It is expected that this new method will find extensive applications
in numerical seismic modelling on complex geological models, and seismic inversion problems.
30
Figure 2.3: Snapshots of z-section of wavefields computed by the new 4th-order compactmethod at (a) t = 0.2s; (b) t = 0.3s; (c) t = 0.4s; (d) t = 0.5s.
31
Chapter 3
An Efficient and High Accuracy
Finite Difference Scheme for Acoustic
Wave Equations
The main content of this part is taken from [37].
3.1 The Compact High-order Scheme
In this paper the 3D acoustic wave equation is considered
utt = ν2(x, y, z)∆u+ s(t, x, y, z), (t, x, y, z) ∈ [0, T ]× Ω (3.1)
with initial conditions and Dirichlet boundary conditions. Here ν(x, y, z) is the acoustic
velocity, s(t, x, y, z) the source function and Ω the computational domain.
The key to obtain high-order compact FD scheme for solving equation (3.1) is to ap-
proximate ∆u with high order compact finite difference approximation. Here one uses a
function of single variable, v(x), to demonstrate the idea. Let xi be the i-th grid point in
the domain and vi be the value of the grid function v(xi). In [4, 12] a general scheme of
32
high-order approximation of second derivative was proposed, which leads to the following
so-called combined compact 4th-order difference approximation of the second derivative
a1v′′i−1 + a0v
′′i + a1v
′′i+1 =
1
h2x
(b1vi−1 + b0vi + b1vi+1
)(3.2)
where a0 = 1, a1 = 110
, b0 = −125
, b1 = 65, and hx is the grid size in x-direction. The 4th-
order accuracy of (3.2) can be easily verified by Taylor expansion. In this paper the idea of
combined compact finite difference scheme will be extended to acoustic wave equations with
variable velocity.
For the sake of simplicity, assume that Ω is a 3D rectangular box defined as
Ω = [xmin, xmax]× [ymin, ymax]× [zmin, zmax],
which is discretized into an (Nx + 2) × (Ny + 2) × (Nz + 2) grid with spatial grid sizes
hx =xmax − xminNx + 1
, hy =ymax − yminNy + 1
and hz =zmax − zminNz + 1
. Then the initial-boundary
value problem of the 3D acoustic wave equation can be rewritten in this form
utt = ν2(x, y, z)∆u+ s(t, x, y, z),
u|t=0 = α(x, y, z), ut|t=0 = β(x, y, z),
u|x=xmin = f0(t, y, z), u|x=xmax = f1(t, y, z),
u|y=ymin = g0(t, x, z), u|y=ymax = g1(t, x, z),
u|z=zmin = h0(t, x, y), u|z=zmax = h1(t, x, y).
(3.3)
Denoted by τ the time step, uni,j,k the numerical solution at grid point (xi, yj, zk) =
(xmin + ihx, ymin + jhy, zmin + khz) and time level tn = nτ . The temporal derivative can
be approximated by the standard 2nd-order centered difference scheme. For the spatial
33
derivatives one has the following 4th-order approximation
a1(uxx)ni−1,j,k + a0(uxx)
ni,j,k + a1(uxx)
ni+1,j,k
=1
h2x
(b1u
ni−1,j,k + b0u
ni,j,k + b1u
ni+1,j,k
),
(3.4)
a1(uyy)ni,j−1,k + a0(uyy)
ni,j,k + a1(uyy)
ni,j+1,k
=1
h2y
(b1u
ni,j−1,k + b0u
ni,j,k + b1u
ni,j+1,k
),
(3.5)
a1(uzz)ni,j,k−1 + a0(uzz)
ni,j,k + a1(uzz)
ni,j,k+1
=1
h2z
(b1u
ni,j,k−1 + b0u
ni,j,k + b1u
ni,j,k+1
),
(3.6)
for 1 6 i 6 Nx, 1 6 j 6 Ny, 1 6 k 6 Nz where (uxx)ni,j,k, (uyy)
ni,j,k, (uzz)
ni,j,k are the sampling
of the second spatial derivatives of u. Define the following vectors
un∗,j,k =
un1,j,k
un2,j,k...
unNx,j,k
Nx×1
, (uxx)n∗,j,k =
(uxx)n1,j,k
(uxx)n2,j,k
...
(uxx)nNx,j,k
Nx×1
, (3.7)
and also define ui,∗,k, (uyy)i,∗,k, ui,j,∗, (uzz)i,j,∗ in similar way. Then the approximation of uxx
can be written in vector form
Ax(uxx)n∗,j,k + a1q
a,nx =
1
h2x
(Bxu
n∗,j,k + b1q
b,nx
)(3.8)
34
where
Ax =
a0 a1
a1 a0 a1
. . . . . .
a1 a0 a1
a1 a0
Nx×Nx
, Bx =
b0 b1
b1 b0 b1
. . . . . .
b1 b0 b1
b1 b0
Nx×Nx
(3.9)
are tridiagonal matrices, and
qa,nx =
(uxx)n0,j,k
0
...
0
(uxx)nNx+1,j,k
Nx×1
, qb,nx =
un0,j,k
0
...
0
unNx+1,j,k
Nx×1
(3.10)
represent the boundary values. In a similar way one can rewrite the approximation of uyy
and uzz in vector form
Ay(uyy)ni,∗,k + a1q
a,ny =
1
h2y
(Byu
ni,∗,k + b1q
b,ny
), (3.11)
Az(uzz)ni,j,∗ + a1q
a,nz =
1
h2z
(Bzu
ni,j,∗ + b1q
b,nz
). (3.12)
The boundary values of uni,j,k are already known. For the boundary values of uxx, uyy and
uzz, one can obtain them from the equation (3.3). For example, for uxx
(uxx)n0,j,k =
(uttν2− s
ν2− uyy − uzz
)|n0,j,k (3.13)
(uxx)nNx+1,j,k =
(uttν2− s
ν2− uyy − uzz
)|nNx+1,j,k. (3.14)
35
Note that u|x=xmin = f0(t, y, z) and u|x=xmax = f1(t, y, z), then one has
(utt)n0,j,k = (∂2
t u)|x=xmin = ∂2t (u|x=xmin) = (∂2
t f0)(tn, yj, zk) (3.15)
(utt)nNx+1,j,k = (∂2
t u)|x=xmax = ∂2t (u|x=xmax) = (∂2
t f1)(tn, yj, zk) (3.16)
(uyy)n0,j,k = (∂2
yu)|x=xmin = ∂2y(u|x=xmin) = (∂2
yf0)(tn, yj, zk) (3.17)
(uyy)nNx+1,j,k = (∂2
yu)|x=xmax = ∂2y(u|x=xmax) = (∂2
yf1)(tn, yj, zk) (3.18)
(uzz)n0,j,k = (∂2
zu)|x=xmin = ∂2z (u|x=xmin) = (∂2
zf0)(tn, yj, zk) (3.19)
(uzz)nNx+1,j,k = (∂2
zu)|x=xmax = ∂2z (u|x=xmax) = (∂2
zf1)(tn, yj, zk). (3.20)
Substitute the above equations into (3.13)(3.14), one obtains
(uxx)n0,j,k =
1
ν20,j,k
[(∂2t f0)nj,k − sn0,j,k
]− (∂2
yf0)nj,k − (∂2zf0)nj,k (3.21)
(uxx)nNx+1,j,k =
1
ν2Nx+1,j,k
[(∂2t f1)nj,k − snNx+1,j,k
]− (∂2
yf1)nj,k − (∂2zf1)nj,k. (3.22)
The boundary values of uyy and uzz can be obtained in a similar way. Now the linear systems
(3.8), (3.11) and (3.12) can be solved. Note that the matrices Ax, Ay and Az are tridiagonal
matrices, thus the linear systems can be solved by tridiagonal matrix algorithm in O(Nx),
O(Ny) and O(Nz) complexity, respectively. Note that one has to solve Ny×Nz linear systems
for uxx, Nx ×Nz linear systems for uyy and Nx ×Ny linear systems for uzz, thus the overall
complexity is O(NxNyNz) for each time step.
In summary one has
(uxx)n∗,j,k = A−1
x
[1
h2x
(Bxu
n∗,j,k + b1q
b,nx
)− a1q
a,nx
](3.23)
(uyy)ni,∗,k = A−1
y
[1
h2y
(Byu
ni,∗,k + b1q
b,ny
)− a1q
a,ny
](3.24)
36
(uzz)ni,j,∗ = A−1
z
[1
h2z
(Bzu
ni,j,∗ + b1q
b,nz
)− a1q
a,nz
](3.25)
i.e. the values of (uxx)ni,j,k, (uyy)
ni,j,k and (uzz)
ni,j,k for 1 6 i 6 Nx, 1 6 j 6 Ny and 1 6 k 6 Nz.
The equation (3.3) also yields
(∂2t u)|t=0 = ν2∆u|t=0 + s|t=0 = ν2∆α + s|t=0, (3.26)
(∂3t u)|t=0 = ν2∆(∂tu)|t=0 + (∂ts)|t=0 = ν2∆β + (∂ts)|t=0, (3.27)
where α = u|t=0 and β = ut|t=0, then one has
(utt)0i,j,k = ν2
i,j,k(∆α)i,j,k + sni,j,k, (3.28)
(uttt)0i,j,k = ν2
i,j,k(∆β)i,j,k + (∂ts)0i,j,k. (3.29)
Thus u−1i,j,k, which denotes the numerical approximation at the ghost time level t = −τ , can
be approximated by
u−1i,j,k =u0
i,j,k − τ(ut)0i,j,k +
1
2τ 2(utt)
0i,j,k −
1
6τ 3(uttt)
0i,j,k +O(τ 4)
=αi,j,k − τβi,j,k +1
2τ 2[ν2
i,j,k(∆α)i,j,k + s0i,j,k]
−1
6τ 3[ν2
i,j,k(∆β)i,j,k + (∂ts)0i,j,k] +O(τ 4).
(3.30)
Finally, the following schemes with error O(τ 2) +O(h4x) +O(h4
y) +O(h4z) is obtained
un+1i,j,k = τ 2[ν2
i,j,k(∆u)ni,j,k + sni,j,k] + 2uni,j,k − un−1i,j,k , n = 0, 1, 2, · · · (3.31)
with u−1i,j,k from (3.30) and (∆u)ni,j,k from (3.23)(3.24)(3.25). A complete scheme with details
is included in Appendix B.1.
Remark 3.1. Note that if there exists two layers of boundary conditions, then an analogy of
(3.2) can lead to 8th-order spatial accuracy [12].
37
3.2 Stability Analysis
The stability analysis employs an energy method which is valid in both constant and variable
coefficients cases. Consider the acoustic equation with zero boundary conditions and zero
source term
utt = ν2(x, y, z)∆u. (3.32)
For simplicity assume hx = hy = hz = h and Nx = Ny = Nz = N . Recall the 4th-order
spatial approximation
a1(uxx)ni−1,j,k + a0(uxx)
ni,j,k + a1(uxx)
ni+1,j,k
=1
h2
(b1u
ni−1,j,k + b0u
ni,j,k + b1u
ni+1,j,k
) (3.33)
a1(uyy)ni,j−1,k + a0(uyy)
ni,j,k + a1(uyy)
ni,j+1,k
=1
h2
(b1u
ni,j−1,k + b0u
ni,j,k + b1u
ni,j+1,k
) (3.34)
a1(uzz)ni,j,k−1 + a0(uzz)
ni,j,k + a1(uzz)
ni,j,k+1
=1
h2
(b1u
ni,j,k−1 + b0u
ni,j,k + b1u
ni,j,k+1
).
(3.35)
Let
A =
a0 a1
a1 a0 a1
. . . . . .
a1 a0 a1
a1 a0
N×N
, B =
b0 b1
b1 b0 b1
. . . . . .
b1 b0 b1
b1 b0
N×N
(3.36)
and Un be the vector form of the numerical solution uni,j,k
Un =
(un1,1,1 . . . unN,1,1 un1,2,1 . . . unN,2,1 un1,3,1 . . . unN,3,1 . . .
)T(3.37)
38
i.e. Un is an N3×1 vector in which uni,j,k is located at the (kN2 +jN+ i)-th row. Also define
Unxx, U
nyy and Un
zz as the vector forms of the second derivatives in a similar way. Then it is
straightforward to see that the equations (3.33)(3.34)(3.35) can be written in the following
form
(A⊗ IN ⊗ IN)Unxx = (B ⊗ IN ⊗ IN)Un (3.38)
(IN ⊗ A⊗ IN)Unyy = (IN ⊗B ⊗ IN)Un (3.39)
(IN ⊗ IN ⊗ A)Unzz = (IN ⊗ IN ⊗B)Un (3.40)
where ⊗ indicates Kronecker product and IN is the N ×N identity matrix. Then one has
Unxx = (A⊗ IN ⊗ IN)−1(B ⊗ IN ⊗ IN)Un, (3.41)
Unyy = (IN ⊗ A⊗ IN)−1(IN ⊗B ⊗ IN)Un, (3.42)
Unzz = (IN ⊗ IN ⊗ A)−1(IN ⊗ IN ⊗B)Un. (3.43)
Lemma 3.2. Kronecker product is associative. The following identities hold
Im ⊗ In = Imn
(A⊗B)(C ⊗D) = (AC)⊗ (BD)
(A⊗B)−1 = (A−1 ⊗B−1)
(A⊗B)T = (AT ⊗BT )
(3.44)
Lemma 3.3. Any eigenvalue of A⊗B arises as a product of eigenvalues of A and B.
39
The above two lemmas can be found in [9]. By those lemmas, one has
A1 : = (A⊗ IN ⊗ IN)−1(B ⊗ IN ⊗ IN) = (A−1B ⊗ IN ⊗ IN)
A2 : = (IN ⊗ A⊗ IN)−1(B ⊗ IN ⊗ IN) = (IN ⊗ A−1B ⊗ IN)
A3 : = (IN ⊗ IN ⊗ A)−1(B ⊗ IN ⊗ IN) = (IN ⊗ IN ⊗ A−1B)
(3.45)
and
σ(A1) = σ(A2) = σ(A3) = σ(A−1B)
where σ denotes the spectrum.
Let denote the entrywise product of matrices, i.e. if A = (aml)q×q and B = (bml)q×q are
two matrices then
A B = (amlbml)q×q. (3.46)
Also denoted by δ2t , the second order temporal finite difference operator,
δ2tU
n = Un+1 − 2Un + Un−1,
note that the time step τ is not included in this operator. Now one can write the numerical
scheme in Section 3.1 for acoustic equations with zero boundary conditions and zero source
term in the following form
δ2tU
n =τ 2
h2· C (A1U
n + A2Un + A3U
n) (3.47)
where C is the vector form of ν2i,j,k. The scheme (3.47) is equivalent to the following form
Φ δ2tU
n = (A1Un + A2U
n + A3Un) := LUn (3.48)
where Φ is the vector form of h2
τ21
ν2i,j,k.
In order to prove the stability of the scheme (3.48), it is necessary to estimate the spec-
40
trum of L = A1 + A2 + A3.
Lemma 3.4. A1, A2 and A3 defined above are self-adjoint matrices, and one has
σ(A1) = σ(A2) = σ(A3) = σ(A−1B) ⊂ (−6,−r(N)] (3.49)
and L defined above is self-adjoint with
σ(L) ⊂ (−18,−3r(N)]. (3.50)
Here
r(N) =
(1 +
1
5cos
π
N + 1
)−1
·(
12
5− 12
5cos
π
N + 1
)> 0. (3.51)
Proof. Note that both A and B are symmetric tridiagonal Toeplitz matrices, then they share
common eigenvectors, see [19, 22]. Also note that A has distinct eigenvalues, as well as B
does. On the other hand, A−1 has the same eigenvectors as A. Thus A−1 and B commute
because of common eigenvectors and distinct eigenvalues. Also note that A−1 is symmetric
since A is. Thus A−1 and B are commutative and both symmetric. Then A−1B is symmetric.
Thus A1, A2 and A3 are all self-adjoint by Lemma 3.2.
The spectra of A and B are given by
σ(A) =
1 +
1
5cos
(πl
N + 1
), l = 1, . . . , N (3.52)
σ(B) =
−12
5+
12
5cos
(πl
N + 1
), l = 1, . . . , N (3.53)
Then one can estimate the spectrum of A−1B by (see [9])
σ(A−1B) ⊂
[−
125
+ 125
cos πN+1
1− 15
cos πN+1
,−125− 12
5cos π
N+1
1 + 15
cos πN+1
]⊂ (−6,−r(N)]. (3.54)
41
Then by Lemma 3.3, one has
σ(A1) = σ(A2) = σ(A3) = σ(A−1B) ⊂ (−6,−r(N)]. (3.55)
Finally L is self-adjoint since it is a sum of self-adjoint matrices and the spectrum of L is
estimated by (see [11])
σ(L) ⊂ (−18,−3r(N)]. (3.56)
Thus a coercive condition for the operator L is obtained
0 < m = 3r(N) 6 −L 6 18 = M. (3.57)
Now one can obtain the main result on the stability.
Theorem 3.5. The new scheme in Section 3.1 is stable if
max16i,j,k6N
νi,j,k · τh
<
√2
3. (3.58)
Proof. The proof is inspired by [2]. Denoted by ‖ · ‖ the l2 norm, and 〈·, ·〉 the l2 inner
product. From the above discussion, the new scheme is equivalent to
Φ δ2tU
n = LUn (3.59)
where Φ is the vector form of h2
τ21
ν2i,j,k.
Let Γn = Un−Un−1, then δ2tU
n = Γn+1−Γn. Now the equation (3.59) can be written as
Φ (Γn+1 − Γn) = LUn. (3.60)
42
Consider the L2 inner product of both sides of (3.60) with Un+1 − Un−1 = Γn+1 + Γn
〈Φ (Γn+1 − Γn), Un+1 − Un−1〉 = 〈LUn, Un+1 − Un−1〉. (3.61)
Since L is self-adjoint, the right-hand-side of (3.61) can be expanded as
〈LUn, Un+1 − Un−1〉 =1
4
[〈LΓn,Γn〉 − 〈L(Un + Un−1), (Un + Un−1)〉
]−1
4
[〈LΓn+1,Γn+1〉 − 〈L(Un+1 + Un), (Un+1 + Un)〉
].
(3.62)
For the left-hand-side of (3.61) one has
〈Φ (Γn+1 − Γn), Un+1 − Un−1〉
=〈Φ (Γn+1 − Γn),Γn+1 + Γn〉
=〈Φ Γn+1,Γn+1〉 − 〈Φ Γn,Γn〉.
(3.63)
Let
Rn = 〈Φ Γn,Γn〉+1
4〈LΓn,Γn〉 − 1
4〈L(Un + Un−1), (Un + Un−1)〉 (3.64)
then (3.61)(3.63)(3.62) result in the identity
Rn+1 = Rn. (3.65)
Note the coercive condition of L in (3.57), then one has
Rn > Φmin‖Γn‖2 − M
4‖Γn‖2 +
m
4‖Un + Un−1‖2, (3.66)
and
Rn 6 Φmax‖Γn‖2 − m
4‖Γn‖2 +
M
4‖Un + Un−1‖2. (3.67)
where Φmax and Φmin are the maximum and minimum entries of the matrix Φ. Recall that
43
all the entries of Φ are positive. Thus if
Φmin −M
4> 0 (3.68)
then Rn is equivalent to the energy given by
‖Γn‖2 + ‖Un + Un−1‖2 = 2‖Un‖2 + 2‖Un−1‖2. (3.69)
Since
Φmin =h2
τ 2· 1
(ν2i,j,k)max
(3.70)
then (3.68) is equivalent to
maxi,j,k
ν2i,j,k ·
τ 2
h2<
4
M=
4
18(3.71)
i.e.
maxi,j,k
νi,j,k ·τ
h<
√2
3. (3.72)
In this case, denoted by en the error at time step tn, the above stability analysis shows that
the energy of the error ‖en‖2 + ‖en−1‖2 conserves during the solving process, which means
the scheme is stable if (3.72) is satisfied.
Remark 3.6. This theorem gives a sufficient condition for stability. The Example 3.4.1
implies that the condition is also necessary.
Remark 3.7. The scheme
1
h2
(6
5vi−1 −
12
5vi +
6
5vi+1
)=
1
10v′′i−1 + v′′i +
1
10v′′i+1
can be written as
1
h2δ2xv =
(1 +
1
12
)δ2xv′′
44
which is formally
1
h2
δ2x
1 + 112δ2x
v = v′′.
Thus the result of the stability analysis in this section is consistent with the result in [14].
3.3 Higher-Order Temporal Accuracy
The new scheme in Section 3.1 is 4th-order accurate in space. However, the total error is
only of 2nd-order in time. In general, it is desirable to get higher order accuracy in time as
well [3]. This section reviews several methods that can improve the temporal accuracy from
2nd-order O(τ 2) to 4th-order O(τ 4).
Pade approximation replaces the 2nd-order centered difference in time 1τ2δ2t by the 4th-
order approximation 1τ2· δ2t
1+ 112δ2t
, then the scheme becomes
δ2t
1 + 112δ2t
uni,j,k = τ 2[ν2i,j,k(∆u)ni,j,k + sni,j,k]. (3.73)
Multiply the both sides of (3.73) by 1 + 112δ2t , one obtains
δ2t u
ni,j,k = τ 2[ν2
i,j,k(1 +1
12δ2t )(∆u)ni,j,k + (1 +
1
12δ2t )s
ni,j,k]. (3.74)
It has been proved in [14] that the scheme (3.74) has a slightly better CFL constant, which
comes from the fact that Pade approximation in time 1τ2· δ2t
1+ 112δ2t
improves the constant 4M
in (3.71) to 6M
which leads to a CFL constant√
33
. However, the trade off is that the scheme
(3.74) is implicit which is expensive to solve for un+1i,j,k directly. An iterative method is needed
there to solve the large sparse linear system.
One can also apply Richardson Extrapolation to improve the temporal accuracy. Denoted
by NS(T ; τ, h), the numerical solution of (3.3) solved by the new scheme with grid size h
and time step τ , evaluated at t = T . Then the Richardson Extrapolation of the numerical
45
solution evaluated at t = T is given by
RE(T ; τ, h) =2lNS(T ; 1
2τ, h)−NS(T ; τ, h)
2l − 1, l = 2, 3, 4, · · · . (3.75)
The Runge-Kutta Method is a different approach which works only for a system of first
order ordinary differential equations. Therefore, one has to rewrite the equation (3.3) as a
first-order system. Then apply explicit Runge-Kutta method to obtain
Un+1 = Un + τ
∑ql=1 blKl
Kl = A(Un + τ
∑q−1p=1 alpKp
)+ S(tn + clτ)
(3.76)
with
U =
u
ut
, A =
0 1
ν2∆ 0
, S =
0
s(t, x, y, z)
, Kl =
Kl,1
Kl,2
(3.77)
and some constants alp, bl, cl. Note that those constants are different from what appear in
Section 3.1.
Here ∆Kl,1 is approximated by the same method as those defined by (3.23), (3.24) and
(3.25) to ensure 4th-order accuracy in space. Thus the boundary value Kl−1|Ω is necessary,
which can be obtained by the same method used for (∆u)|Ω in (3.13), see Appendix B.2.
One numerical example has been solved to show that both Richardson extrapolation and
Runge-Kutta methods can improve the order of accuracy to 4th-order in Section 3.4. It is
worthy to mention that extra caution should be taken when the two methods are used, as
they might bring in some stability issues.
Remark 3.8. The stability analysis in Section 3.2 no longer applies in the Richard Extrap-
olation case and the Runge-Kutta case. However, the numerical experiments in Section 3.4
do not show any stability problem in those cases.
46
3.4 Numerical Experiments
In this section, four numerical experiments are conducted with the new scheme to demon-
strate the efficiency and accuracy. The first example verifies the 2nd-order in time and
4th-order in space accuracy. The second example is used to further validate the order of
accuracy of the new method. The third example compares the performance of Richardson
Extrapolation and Runge-Kutta Methods in improving temporal accuracy to 4th-order. The
fourth example considers a more realistic problem, solving an underground acoustic model
with Ricker wavelet source.
3.4.1 Example 1
This example solves the acoustic wave equation defined on the domain Ω = [0, 1]×[0, 1]×[0, 1]
and t ∈ [0, 1]
utt = ν2∆u+ s, (3.78)
where
ν2 =1
(x− 12)(y − 1
2)(z − 1
2) + 1
6
(3.79)
and
s = (4− 14ν2)e2tex+2y+3z (3.80)
with initial and Dirichlet boundary conditions compatible to the analytical solution which
is given by
u = e2tex+2y+3z. (3.81)
In order to validate that the new scheme is 2nd-order in time and 4th-order in space, let
hx = hy = hz = h, and set the grid size h and time step τ to satisfy τ = h2. The errors
in max norm and energy norm with different h are listed in Table 3.1, which show that the
47
Table 3.1: Numerical errors in max norm and energy norm in Example 3.4.1 with τ = h2.Note that the CFL condition requires τ
h<√
318
.
h 1/15 1/20 1/25 1/30Emax 9.5748e-04 3.0609e-04 1.2605e-04 6.1007e-05Eenergy 3.8709e-04 1.1990e-04 4.8467e-05 2.3163e-05
Conv. Order (max) - 3.9642 3.9759 3.9803Conv. Order (energy) - 4.0739 4.0592 4.0496
new scheme is indeed of error O(τ 2) +O(h4). Here the accuracy order is calculated by
Conv. Order =log[E(h1)/E(h2)
]log(h1/h2)
. (3.82)
Remark 3.9. Since τ = h2 in this case, it is actually verified that the new scheme is of accuracy
O(τ 2) + O(h4) = O(h4). Note that the number of time steps is of order O(τ−1) = O(h−2),
and the approximation of u(−τ) is of order O(τ 4) = O(h8), see (3.30), thus the aggregate
error from u(−τ) is of order O(h−2h8) = O(h6) O(h4).
The Example 3.4.1 also can be used to numerically validate the estimated CFL condition
given by Theorem 3.5. To this end, choosing τ and h such that τh
is slightly greater than
νmax√
23
, so that the stability condition is violated. From (3.79), one can derive that νmax =√
24. Therefore, the CFL condition requires τh< 1
6√
3≈ 0.0962. To slightly violate the CFL
condition, choosing τ = 1200
and h = 130
, which gives the ratio τh
= 0.15. In this case, the
numerical errors are Emax = 4.8304×1014 and Eenergy = 1.0085×1013, respectively. Clearly,
the numerical solution is not convergent. To further show that the numerical solution will
fail to converge when the CFL condition is violated, a long time case for the same example
was run. Here choosing τ = 1200
and h = 120
, so the ratio τh
= 0.1 is just slightly above
the CFL condition. Solving the equation from t = 0 to t = 5, the maximal norm of the
numerical error is Emax = 8.8861 × 106. It is clear that the numerical solution is divergent
when the CFL condition is not satisfied.
48
Table 3.2: Numerical errors in max norm and energy norm in Example 3.4.2 with τ = h2.Note that the CFL condition requires τ
h<√
69
.
h 1/10 1/15 1/20 1/25Emax 1.1904e-05 2.5583e-06 8.1904e-07 3.3785e-07Eenergy 3.4651e-06 6.7639e-07 2.1147e-07 8.5840e-08
Conv. Order (max) - 3.7920 3.9591 3.9684Conv. Order (energy) - 4.0293 4.0416 4.0404
3.4.2 Example 2
Here is another example to validate the accuracy order of the new scheme. Consider the
acoustic wave equation defined on the domain Ω = [0, 1]× [0, 1]× [0, 1] and t ∈ [0, T ]
utt = ν2∆u+ s
u|t=0 = sin(πx) sin(πy) sin(πz)
ut|t=0 = 0
u|∂Ω = 0
(3.83)
with
ν2 = 1 + xy + z (3.84)
and
s =[−1 + 3π2(1 + xy + z)
]cos(t) sin(πx) sin(πy) sin(πz). (3.85)
The analytic solution of this equation is
u = cos(t) sin(πx) sin(πy) sin(πz). (3.86)
Again let hx = hy = hz = h, and set the grid size h and time step τ to satisfy τ = h2. The
errors in max norm and energy norm with different h are listed in Table 3.2. The accuracy
order is again validated.
49
3.4.3 Example 3
This example compares Richardson Extrapolation (RE) and Runge-Kutta Method (RK) in
increasing temporal accuracy to 4th-order. Consider the 3D acoustic wave equation defined
on Ω = [0, 1]× [0, 1]× [0, 1] and t ∈ [0, T ].,
utt = (1 + xyz)∆u+ s(t, x, y, z)
u|t=0 = sin(πx) sin(πy) sin(πz)
ut|t=0 = π sin(πx) sin(πy) sin(πz)
u|∂Ω = 0
(3.87)
with source
s(t, x, y, z) = (4 + 3xyz)π2eπt sin(πx) sin(πy) sin(πz). (3.88)
The analytical solution is given by
u = eπt sin(πx) sin(πy) sin(πz). (3.89)
For simplicity, let hx = hy = hz = h and τ = h10
in all test cases. Denoted by NS(T ; τ, h),
the numerical solution of (3.87) solved by the new scheme with grid size h and time step τ ,
evaluated at t = T . Then the Richardson Extrapolation of the numerical solution evaluated
at t = T is given by
RE(T ; τ, h) =4NS(T ; 1
2τ, h)−NS(T ; τ, h)
3. (3.90)
Rewrite the equation (3.87) as a first-order system
Ut = AU + S(t) (3.91)
50
Table 3.3: Comparison of RE and RK4 in Example 3.4.3 in energy norm, with τ = h10
.
h 1/10 1/15 1/20 1/25ERE 2.9340e-04 5.4765e-05 1.6862e-05 6.7968e-06ERK4 2.9327e-04 5.4734e-05 1.6852e-05 6.7924e-06
RE Conv. Order - 4.1397 4.0948 4.0719RK4 Conv. Order - 4.1400 4.0949 4.0721RE CPU time (s) 0.175715 0.674665 1.678161 3.572069
RK4 CPU time (s) 0.271528 0.966194 2.412038 5.024197
with
U =
u
ut
, A =
0 1
(1 + xyz)∆ 0
, S(t) =
0
s(t)
, Kl =
Kl,1
Kl,2
. (3.92)
Then one can consider the conventional 4th-order Runge-Kutta Method (RK4)
Un+1 = Un + τ6
(K1 + 1
2K2 + 1
2K3 +K4
)K1 = AUn + S(tn)
K2 = A(Un + τ
2K1
)+ S(tn + 1
2τ)
K3 = A(Un + τ
2K2
)+ S(tn + 1
2τ)
K4 = A (Un + τK3) + S(tn + τ)
. (3.93)
Both RE and RK4 should improve the temporal accuracy from O(τ 2) to O(τ 4), which leads
to an overall 4th-order temporal and spatial accuracy. Denoted by ERE the error of RE, and
ERK4 the error of RK4, Table 3.3 compares those two methods in energy norm. From Table
3.3 one can see that RK4 has slightly better accuracy than RE, but it is more time expensive.
The reason is stated below. Comparing to the original scheme RE does an additional solving
process for τ2, which leads to an overall computational cost about 3 times as much as the
original one. However, RK4 used here requires additionally approximating four ∆Kl’s in
each time step which leads to an overall computational cost about 5 times as much as the
51
original one.
3.4.4 Example 4
This example solves a more realistic problem in which the seismic wave is generated by
a Ricker wavelet source. The region is a three-dimensional domain Ω = [0m, 1200m] ×
[0m, 1200m]× [0m, 1350m]. The velocity is given by
ν(x, y, z) =
1200m/s, if 0 6 z 6 879.75
2500m/s, if 879.75 < z 6 1350
. (3.94)
This can be regarded as that the region is divided in two parts. From the ground surface to
879.75m underground is soil with sound speed ν = 1200m/s, and from 879.75m underground
to 1350m underground is rock with sound speed ν = 2500m/s. The underground model is
sketched in Figure 3.1. The Ricker wavelet source is given by
s(t, x, y, z) = δ(x− xs, y − ys, z − zs)[1− 2π2f 2p (t− dr)2]e−π
2f2p (t−dr)2 (3.95)
with dominant frequency fp = 10Hz, temporal delay dr = 0.5/fp. The wave generator
is located at (xs, ys, zs) = (600m, 600m, 600m), which is in the soil area. The time step
τ = 5 × 10−4s and grid size h = 5m are chosen to satisfy Nyquist Theorem on spatial
resolution and the stability condition given in (3.72)
νmaxτ
h<
√2
3. (3.96)
Four snapshots of y-section at y = ys are plotted. Note that the distance between the
source and the rock area is 879.75m − 600m = 279.75m, thus the wave will reach the rock
area at t = 279.751200
s = 0.233125s. Figure 3.2 shows that at t = 0.225s the y-section of the
52
Figure 3.1: Underground model for Example 3.4.4, the yellow part is soil with sound speedν = 1200m/s, the gray part is rock with sound speed ν = 2500m/s. The wave generator islocated in soil area.
wave is still a circle. Figure 3.3 shows that at t = 0.375s the wave has reached the rock
area, thus both reflection and refraction occur when the wave crosses the interface of the
two media at z = 897.75m. Figure 3.4 shows the wave has been reflected back from bottom
boundary before the snap time t = 0.42s. Figure 3.5 shows the reflected waves at t = 0.66s.
Note that on Figure 3.5, the wave reflected from bottom boundary has crossed the interface
of soil and rock, thus it shows a different shape from Figure 3.4 due to refraction.
There are also some spurious waves propagating faster than the true wave on those
figures. One may have to zoom in the figures to find them. They are inconspicuous and
neglectable compared to the true wave. Their occurrence is resulted from the way how ∆u
is approximated in this new method. For simplicity consider an interval Ω = [−2, 2] and a
smooth function v supported on [−1, 1]. The second derivative on Ω is approximated by
vxx = A−1Bv (3.97)
where A and B are given by (3.36). The matrix A−1 is full, thus it will spread a little part
53
Figure 3.2: Snapshot of y-section at t = 0.225s in Example 3.4.4
Table 3.4: Comparison of the reference solution and numerical solutions in Example 3.4.4with τ = 0.5× 10−4s.
h (m) 25 15 10 7.5Eenergy 1.3216e-09 8.7131e-11 8.8250e-12 1.6799e-12Emax 4.5621e-06 7.0167e-07 1.4546e-07 4.4025e-08
Conv. Order (energy) - 5.3232 5.6474 5.7662Conv. Order (max) - 3.6648 3.8808 4.1545
of v to its second derivative outside of the support [−1, 1], which makes the approximation
of vxx non-zero on [−2,−1] and [1, 2].
A reference solution for this example is computed to validate the accuracy order of the
new scheme. The reference solution comes with h = 2.5m and τ = 0.5 × 10−4s, computed
from initial time t = 0s to end time t = 0.3s. Comparing it with numerical solutions with
τ = 0.5× 10−4s and h = 25m, 15m, 10m, 7.5m, the results are shown in Table 3.4.
It is observed that the accuracy orders in energy norm are greater than four. One of the
possible reasons may be that the reference solution and the numerical solutions are computed
by the same method, thus there may be errors with the same pattern and those errors cancel
54
Figure 3.3: Snapshot of y-section at t = 0.375s in Example 3.4.4
out themselves.
3.5 Conclusion
In this Chapter a compact explicit 2nd-order in time and 4th-order in space FDM has been
developed to solve acoustic equations with variable acoustic velocity. The new scheme has a
time complexity which is linear to the total number of grid points for each time step. One
feature of this new method is that a wave equation based analytical approach is developed
to approximate the boundary conditions for the second spatial derivatives uxx, uyy and
uzz so that the combined compact scheme can be applied efficiently. This new scheme is
conditionally stable with a slightly lower CFL constant than approximating the temporal
2nd derivative by Pade approximation which makes the scheme implicit. Three numerical
examples for which the analytical solutions are available were solved by the new scheme
to validate the accuracy order and the compatibility with Richardson Extrapolation and
4th-order Runge-Kutta Method. A more realistic problem has been solved to show that the
55
Figure 3.4: Snapshot of y-section at t = 0.42s in Example 3.4.4
method is accurate and efficient for numerical simulation of acoustic wave propagation in 3D
heterogeneous media. The new scheme is expected to find wide applications in numerical
seismic modeling and related areas.
56
Figure 3.5: Snapshot of y-section at t = 0.66s in Example 3.4.4
57
Chapter 4
Efficient and Stable Finite Difference
Modeling of Acoustic Wave
Propagation in Variable Density
Media
The main content of this part is taken from [36].
4.1 The New Compact Scheme
In this part, the following 3D acoustic wave equation in heterogeneous media with initial
condition and Dirichlet boundary conditions is considered,
1
ρc2utt −∇ ·
(1
ρ∇u)
= s, (t, x, y, z) ∈ [0, T ]× Ω, (4.1)
where s is the source term, c the acoustic velocity and ρ the media density. Here s is a
function of x, y, z and t, while c and ρ are functions of x, y and z. The solution u represents
the acoustic pressure.
58
Note that
∇ ·(
1
ρ∇u)
= ∂x
(1
ρ∂xu
)+ ∂y
(1
ρ∂yu
)+ ∂z
(1
ρ∂zu
), (4.2)
it is necessary to approximate each term of the right-hand side with 4th-order accuracy to
obtain an overall 4th-order spatial accuracy.
Consider a single-variable function, v(x). In [12, 4], the authors proposed the so-called
combined compact 4th-order finite difference approximation for the first derivative,
1
4v′i−1 + v′i +
1
4v′i+1 =
1
h
(−3
4vi−1 +
3
4vi+1
)(4.3)
where vi is the function v evaluated at the grid point xi, which is v(xi), h the grid size.
Similarly, the second derivative can be approximated in 4th-order by
1
10v′′i−1 + v′′i +
1
10v′′i+1 =
1
h2
(6
5vi−1 −
12
5vi +
6
5vi+1
). (4.4)
The 4th-order accuracy of the scheme (4.3)(4.4) can be verified by Taylor expansion.
For simplicity, assume that Ω is a rectangle box defined as
Ω = [xmin, xmax]× [ymin, ymax]× [zmin, zmax],
which is discretized into an (Nx + 2) × (Ny + 2) × (Nz + 2) grid with spatial grid sizes
hx =xmax − xminNx + 1
, hy =ymax − yminNy + 1
and hz =zmax − zminNz + 1
. Then the initial-boundary
59
value problem of the 3D acoustic wave equation can be rewritten in this form
1ρ(x,y,z)c2(x,y,z)
utt −∇ ·(
1ρ(x,y,z)
∇u)
= s(t, x, y, z),
u|t=0 = α(x, y, z), ut|t=0 = β(x, y, z),
u|x=xmin = f0(t, y, z), u|x=xmax = f1(t, y, z),
u|y=ymin = g0(t, x, z), u|y=ymax = g1(t, x, z),
u|z=zmin = h0(t, x, y), u|z=zmax = h1(t, x, y).
(4.5)
Denoted by τ the time step, uni,j,k the numerical solution at grid point (xi, yj, zk) = (xmin +
ihx, ymin + jhy, zmin + khz) and time level tn = nτ , also define ρi,j,k, ci,j,k and sni,j,k in similar
way. Define the following vectors
un∗,j,k =
un1,j,k
un2,j,k...
unNx,j,k
Nx×1
, (ux)n∗,j,k =
(ux)n1,j,k
(ux)n2,j,k
...
(ux)nNx,j,k
Nx×1
(4.6)
and
[(1
ρux
)x
]n∗,j,k
=
[(1ρux
)x
]n1,j,k[(
1ρux
)x
]n2,j,k
...[(1ρux
)x
]nNx,j,k
Nx×1
, (4.7)
where 1 6 i 6 Nx, 1 6 j 6 Ny, 1 6 k 6 Nz. Throughout the rest of this part, an object
represented by a lower-case letter with ∗ being one of its subscripts denotes the vector form
of the object, which is defined in the similar way as above. Then from (4.3) one has the
60
approximation equations for[(
1ρux
)x
]n∗,j,k
Ax(ux)n∗,j,k +
1
4wa,nx =
1
hx
(Bxu
n∗,j,k +
3
4wb,nx
), (4.8)
and
Ax
[(1
ρux
)x
]n∗,j,k
+1
4wa,nxx =
1
hx
(Bx
[1
ρux
]n∗,j,k
+3
4wb,nxx
), (4.9)
where
Ax =
1 14
14
1 14
. . . . . .
14
1 14
14
1
Nx×Nx
, Bx =
0 34
−34
0 34
· · · · · ·
−34
0 34
−34
0
Nx×Nx
(4.10)
are tridiagonal matrices, and
wa,nx =
(ux)n0,j,k
0
...
0
(ux)nNx+1,j,k
Nx×1
, wb,nx =
−un0,j,k
0
...
0
unNx+1,j,k
Nx×1
(4.11)
wa,nxx =
[(1ρux
)x
]n0,j,k
0
...
0[(1ρux
)x
]nNx+1,j,k
Nx×1
, wb,nxx =
−[
1ρux
]n0,j,k
0
...
0[1ρux
]nNx+1,j,k
Nx×1
(4.12)
represent the boundary values. Note that the negative sign of the first components of wb,nx
and wb,nxx results from the negative term of the right-hand side of (4.3). The approxima-
61
tion equations for
[(1ρuy
)y
]ni,∗,k
and[(
1ρuz
)z
]ni,j,∗
can be obtained similarly. Note that for
the boundary values vectors wx’s and wxx’s, only wb,nx can be evaluated directly from the
boundary conditions of the equation. For the boundary values occur in wa,nx , wb,nxx and wa,nxx ,
they can be obtained by the 4th-order one-sided finite difference approximation for the first
derivatives,
v′0 =1
hx
(−25
12v0 + 4v1 − 3v2 +
4
3v3 −
1
4v4
), (4.13)
and
v′Nx+1 =1
hx
(25
12vNx+1 − 4vNx + 3vNx−1 −
4
3vNx−2 +
1
4vNx−3
). (4.14)
Suppose that un∗,j,k is known, the following steps show how to obtain[(
1ρux
)x
]n∗,j,k
from the
above discussion.
1. Both un0,j,k and unNx+1,j,k are evaluated from the boundary conditions of the equation,
thus wb,nx can be obtained.
2. Use (4.13)(4.14) to approximate (ux)n0,j,k and (ux)
nNx+1,j,k by un∗,j,k, thus wa,nx will be
known.
3. Solve (4.8) to obtain (ux)n∗,j,k, then
(1ρux
)n∗,j,k
will be known.
4. Since 1ρ
is known, thus(
1ρux
)n0,j,k
and(
1ρux
)nNx+1,j,k
will also be known from Step 2.
5. Use (4.13)(4.14) to approximate[(
1ρux
)x
]n0,j,k
and[(
1ρux
)x
]nNx+1,j,k
by(
1ρux
)n0,j,k
,(1ρux
)nNx+1,j,k
and(
1ρux
)n∗,j,k
, thus wa,nxx will be known.
6. Solve (4.9) to obtain[(1ρux)x
]n∗,j,k
.
The assumption that un∗,j,k is known is reasonable, since this part uses the conventional 2nd-
order central finite difference to approximate the second derivative utt, i.e. un+1 is solved
from un and un−1. The derivative terms
[(1ρuy
)y
]ni,∗,k
and[(
1ρuz
)z
]ni,j,∗
can be obtained
similarly.
62
Remark 4.1. If the media density ρ is differentiable, then one can also consider the equivalent
form of the Laplacian ∇ · (1ρ∇u) = ∇1
ρ· ∇u + 1
ρ∆u, which will be simpler to implement,
where ∇u is approximated by the 4th-order compact finite difference scheme as above, and
∆u can be approximated by the 4th-order compact finite difference in [37]. Note that the
approximation of the second spatial derivatives of u on the boundary requires a one-sided
finite difference approximation similar to (4.13) and (4.14) with different coefficients.
To initialize the solving process, one also needs u−1i,j,k = u(−τ, xi, yj, zk), which can be
obtained by
u−1i,j,k =u0
i,j,k − τ(ut)0i,j,k +
1
2τ 2(utt)
0i,j,k −
1
6τ 3(uttt)
0i,j,k +O(τ 4)
=αi,j,k − τβi,j,k +1
2τ 2
(ρc2)i,j,k
[∇ ·(
1
ρ∇α)]
i,j,k
+ s0i,j,k
−1
6τ 3
(ρc2)i,j,k
[∇ ·(
1
ρ∇β)]
i,j,k
+ (∂ts)0i,j,k
+O(τ 4)
(4.15)
where α = u|t=0 and β = ut|t=0 are the initial conditions.
Finally, a compact finite difference scheme with error O(τ 2) +O(h4x) +O(h4
y) +O(h4z) is
obtained as
un+1i,j,k = τ 2
(ρc2)i,j,k
[∇ ·(
1
ρ∇u)]n
i,j,k
+ sni,j,k
+ 2uni,j,k − un−1
i,j,k , n = 0, 1, 2, · · · (4.16)
with[∇ ·(
1ρ∇u)]n
i,j,kobtained from
[(1ρux)x
]n∗,j,k
,[(1ρuy)y
]ni,∗,k
and[(1ρuz)z
]ni,j,∗
, and u−1i,j,k
from (4.15).
4.2 Stability Analysis of the New Scheme
Consider the acoustic wave equation with zero boundary conditions and zero source term
1
ρc2utt −∇ ·
(1
ρ∇u)
= 0, (4.17)
63
or
utt − c2ρ
[∇ ·(
1
ρ∇u)]
= 0. (4.18)
For simplicity assume hx = hy = hz = h, Nx = Ny = Nz = N , τ the time step size. Also let
Un be the vector form of the numerical solution uni,j,k denoted by
Un =
(un1,1,1 . . . unN,1,1 un1,2,1 . . . unN,2,1 un1,3,1 . . . unN,3,1 . . .
)T(4.19)
i.e. Un is an N3×1 vector in which uni,j,k is located at the (kN2 +jN+i)-th row. Also denote
D2xU , D2
yU and D2zU as the vector form of the derivative term
[(1ρux
)x
]ni,j,k
,
[(1ρuy
)y
]ni,j,k
and[(
1ρuz
)z
]ni,j,k
, respectively.
Then equation (4.18) can be approximated by
1
τ 2δ2tU
n − C2Q(D2xU
n +D2yU
n +D2zU
n)
= 0, (4.20)
where Q is the diagonal matrix for the function ρ
Q = diag
(ρ1,1,1 . . . ρN,1,1 ρ1,2,1 . . . ρN,2,1 ρ1,3,1 . . . ρN,3,1 . . .
)(4.21)
and C is the diagonal matrix for the function c
C = diag
(c1,1,1 . . . cN,1,1 c1,2,1 . . . cN,2,1 c1,3,1 . . . cN,3,1 . . .
). (4.22)
In other words, Q and C are N3×N3 diagonal matrices whose diagonal entries are the vector
form of qi,j,k and ci,j,k, respectively.
64
Let
A =
1 14
14
1 14
. . . . . .
14
1 14
14
1
N×N
, B =
0 34
−34
0 34
· · · · · ·
−34
0 34
−34
0
N×N
(4.23)
Then the spatial derivative terms D2xU , D2
yU and D2zU of equation (4.20) can be written as
D2xU
n =1
h2
[A−1
1 B1Q−1A−1
1 B1
]Un, (4.24)
D2yU
n =1
h2
[A−1
2 B2Q−1A−1
2 B2
]Un, (4.25)
D2zU
n =1
h2
[A−1
3 B3Q−1A−1
3 B3
]Un, (4.26)
where
A1 = A⊗ IN ⊗ IN , B1 = B ⊗ IN ⊗ IN , (4.27)
A2 = IN ⊗ A⊗ IN , B2 = IN ⊗B ⊗ IN , (4.28)
A3 = IN ⊗ IN ⊗ A, B3 = IN ⊗ IN ⊗B, (4.29)
with ⊗ being the Kronecker product and IN the N ×N identity matrix.
Now define
L = C2Q(A−1
1 B1Q−1A−1
1 B1 + A−12 B2Q
−1A−12 B2 + A−1
3 B3Q−1A−1
3 B3
), (4.30)
then the discretized equation (4.20) can be written as
1
τ 2δ2tU
n − 1
h2LUn = 0, (4.31)
65
where δ2t is the conventional temporal 2nd-order central finite difference operator
δ2tU
n = Un+1 − 2Un + Un−1. (4.32)
It will be very useful if the estimate of the eigenvalues of the spatial difference operator is
known. However, it is very difficult to obtain the estimate for L, due to the variant coefficients
in Q and C. It is empirical that the eigenvalue of L with largest absolute value determines
the stability of a scheme for wave equations. Thus, consider freezing the coefficients of L,
which leads to a variant of L,
L = c2max
ρmaxρmin
(A−1
1 B1A−11 B1 + A−1
2 B2A−12 B2 + A−1
3 B3A−13 B3
), (4.33)
where cmax and ρmax are the maximum entries of C and Q, respectively, ρmin the minimum
entry of Q. Note that all of the entries of C and Q are positive.
4.2.1 Estimation of the Eigenvalues
To estimate the spectrum of L, we introduce the following lemmas, which can be found in
[9].
Lemma 4.2. Kronecker product is associative. For square matrices K, L, G and H, the
following identities hold
IM ⊗ IN = IMN ,
(K ⊗ J)(G⊗H) = (KG)⊗ (JH),
(K ⊗ J)−1 = (K−1 ⊗ J−1),
(K ⊗ J)T = (KT ⊗ JT ).
(4.34)
Lemma 4.3. For square matrices K and J , any eigenvalue of K ⊗ J arises as a product of
eigenvalues of K and J . If λK and λJ are eigenvalues of K and J , respectively, then λK ·λJ
66
is an eigenvalue of K ⊗ J .
With the above lemmas, one has
A−11 B1A
−11 B1 = (A−1BA−1B)⊗ IN ⊗ IN ,
A−12 B2A
−12 B2 = IN ⊗ (A−1BA−1B)⊗ IN ,
A−13 B3A
−13 B3 = IN ⊗ IN ⊗ (A−1BA−1B),
(4.35)
thus
σ(A−1
1 B1A−11 B1
)= σ
(A−1
2 B2A−12 B2
)= σ
(A−1
3 B3A−13 B3
)= σ
(A−1BA−1B
). (4.36)
Recall the definition of Kronecker sum of two square matrices KM×M and LN×N
K ⊕ J = (IN ⊗K) + (J ⊗ IM). (4.37)
The following lemma can also be found in [9].
Lemma 4.4. For square matrices K and J , any eigenvalue of K ⊕ J arises as a sum of
eigenvalues of K and J . If λK and λJ are eigenvalues of K and J , respectively, then λK +λJ
is an eigenvalue of K ⊕ J .
Based on that, we can verify that
K ⊕ J ⊕G = (IN ⊗ IN ⊗K) + (IN ⊗ J ⊗ IN) + (G⊗ IN ⊗ IN) (4.38)
and
σ(K ⊕ J ⊕G) = λK + λJ + λG, λK ∈ σ(K), λJ ∈ σ(J), λG ∈ σ(G). (4.39)
67
With (4.33)(4.35), one has
L = c2max
ρmaxρmin
[(A−1BA−1B)⊕ (A−1BA−1B)⊕ (A−1BA−1B)
]. (4.40)
Now it is necessary to find out the eigenvalues of A−1BA−1B. Since A and B are tridiagonal
Toeplitz matrices, their eigenvalues are given by
σ(A) =
1 +
1
2cos
lπ
N + 1, 1 6 l 6 N
⊂(
1
2,3
2
), (4.41)
and
σ(B) =
3√−1
2cos
lπ
N + 1, 1 6 l 6 N
⊂(−3√−1
2,3√−1
2
). (4.42)
where√−1 takes the principal branch. Therefore, A is a positive definite matrix.
Recall that for a square matrix K, the numerical range of F (K) is defined as
F (K) =
~vTK~v
~vT~v, v non-zero complex vectors
, (4.43)
which is a closed bounded convex set containing all of the eigenvalues of K. Furthermore, if
K is a normal matrix, F (K) is exactly the convex closure of all the eigenvalues of K.
Theorem 4.5. For two real square matrices K and J of the same size, if K is a symmetric
positive semi-definite matrix and λ is an eigenvalue of KJ , then
λ ∈ F (K)F (J) = λK · λJ , λK ∈ F (K) and λJ ∈ F (J), (4.44)
in other words, σ(KJ) ⊂ F (K)F (J).
The theorem above was proven in [41].
Now since A is a symmetric semi-positive definite matrix, so is A−1. Since A−1 is sym-
metric and B is anti-symmetric, thus, they are both normal matrices. Consequently, the
numerical ranges of F (A−1) and F (B) coincide with the closure of all the eigenvalues of A−1
68
and B, respectively, i.e.
F (A−1) =
[2
3, 2
], F (B) =
[−3√−1
2,3√−1
2
]. (4.45)
Then the eigenvalues of A−1B can be estimated by
σ(A−1B) ⊂ F (A−1)F (B) =[−3√−1, 3
√−1]. (4.46)
Then one has
σ(A−1BA−1B) ⊂ [−9, 0] . (4.47)
Finally, by (4.39)(4.40), one has
σ(L) ⊂[−27c2
max
ρmaxρmin
, 0
]. (4.48)
4.2.2 Stability
We now derive an empirical CFL condition for the new scheme. Consider the coefficient-
frozen version of the equation (4.31)
δ2tU
n − τ 2
h2LUn = 0 (4.49)
i.e.
Un+1 −(
2 +τ 2
h2L
)Un + Un−1 = 0. (4.50)
Inspired by von Neumann analysis, one considers the following characteristic equation
λ2 − (2− 27r)λ+ 1 = 0 (4.51)
69
where r = c2max
qmaxqmin
τ2
h2. Here the finite difference operator L is replaced by −27c2
maxqmaxqmin
,
which is the upper bound of the eigenvalue of L with largest absolute value. The scheme is
stable if the roots of the characteristic equation (4.51) satisfy |λ1| 6 1 and |λ2| 6 1. From
the characteristic equation we already have λ1λ2 = 1, thus we only need the two roots to be
complex numbers, i.e. the discriminant should be negative
∆ = (2− 27r)2 − 4 = 27r(27r − 4) < 0, (4.52)
thus one has
0 < r = c2max
ρmaxρmin
τ 2
h2<
4
27, (4.53)
i.e.
cmax
√ρmaxρmin
· τh<
2
3√
3. (4.54)
Theorem 4.6. The new scheme will be stable if the CFL condition (4.54) is satisfied.
4.3 Numerical Experiments
In this section, four numerical examples are solved to demonstrate the accuracy and efficiency
of the new scheme. The first example is solved to validate that the new scheme is of 2nd-
order accuracy in time and 4th-order accuracy in space. The second example solves a zero
initial and boundary conditions problem with Ricker wavelet source. The third example
validates the effectiveness and the accuracy of the new scheme for acoustic wave equation
with PML boundary conditions. The fourth example considers a more realistic problem, in
which the Marmousi 2 model is used in the simulation of the seismic wave propagation.
4.3.1 Example 1
This example validates that the new scheme is of 2nd-order accuracy in time and 4th-order
accuracy in space. Consider the acoustic wave equation defined on the domain [0, 1]× [0, 1]×
70
Table 4.1: Numerical errors in max norm in Example 4.3.1 with τ = h2. Note that the CFLcondition requires τ
h< 0.1156.
h 1/10 1/16 1/20 1/24 1/32τ 1/100 1/256 1/400 1/576 1/1024E 7.6115e-05 9.5211e-06 3.8419e-06 1.7292e-06 5.0288e-07
Conv. Order - 4.4228 4.0671 4.3786 4.2931
[0, 1] and t ∈ [0, 1],
utt − ρc2∇ ·(
1
ρ∇u)
= ρc2s, (4.55)
where
ρ = e(−x−y−z)/3, (4.56)
c2 = 1 +1
2xyz, (4.57)
and
ρc2s =− sin(t) cos(x+ 2y + 3z)
+ sin(t)
(1 +
1
2xyz
)[14 cos(x+ 2y + 3z) + 2 sin(x+ 2y + 3z)]
(4.58)
with initial and boundary conditions compatible to the analytic solution
u = sin(t) cos(x+ 2y + 3z). (4.59)
The grid sizes are given by hx = hy = hz = h and the time step is τ = h2. Thus it is sufficient
to show that in this numerical experiment, the accuracy order is O(τ 2) + O(h4) = O(h4).
The max errors with different h are listed in Table 4.1, which clearly validated the desired
convergence order calculated as
Conv. Order =log[E(h1)/E(h2)
]log(h1/h2)
. (4.60)
71
4.3.2 Example 2
In this example we solve the wave equation with Ricker wavelet source. Consider the equation
on the region Ω = [0, 2]× [0, 2]× [0, 2]
1
ρutt −∇ ·
(1
ρ∇u)
= s, (4.61)
with zero initial and boundary conditions u|t=0 = 0, ut|t=0 = 0 and u|∂Ω = 0, where ρ =
2z2 + 1 and s is the Ricker wavelet source given by
s(t, x, y, z) = δ(x− x0, y − y0, z − z0)[1− 2π2f 2p (t− dr)2]e−π
2f2p (t−dr)2 , (4.62)
with dominant frequency fp = 10 and temporal delay dr = 12fp
. The source is placed at the
centre of the region with x0 = y0 = z0 = 1. The equation is solved by the new scheme with
hx = hy = hz = h = 140
and τ = 1400
. The stability condition which requires τh< 2
9√
3≈ 0.1283
is satisfied.
Three snapshots of y-section at y = 1 at different times are plotted in Figure 4.1, Figure
4.2 and Figure 4.3. Note that the media density is given by ρ = 2z2 + 1, which depends only
on z. Those figures show that the wave propagates at constant sound speed, as expected.
However, the energy is more concentrated in the area with higher media density. Finally, it
is observed that the boundaries reflexes the wave back due to zero boundary conditions.
Remark 4.7. In real-world applications, variable media density usually results in variable
acoustic velocity in different areas of the domain. However, in this example, for the sake of
simplicity, the acoustic velocity is normalized to a constant c = 1 to highlight the effect of
the variable media density on the propagation of the acoustic wave.
Remark 4.8. It is worth to mention that Figure 4.1, Figure 4.2 and Figure 4.3 are y-section
of a wave in 3D space, thus the energy component of the wave in y-direction is not shown
on those figures.
72
Figure 4.1: Snapshot of y-section of the acoustic pressure u at t = 0.4s in Example 4.3.2
Three snapshots of z-section at z = 1 at different time are plotted in Figure 4.4, Figure
4.5 and Figure 4.6. It is clear that the wave propagates as if the density is constant, which
is true as the density is independent of x and y. As can be seen, the energy is distributed
symmetrically about the centre of the z-section. Similarly, it is noticed that the boundaries
reflect the wave back due to zero boundary conditions.
4.3.3 Example 3
Perfectly Matched Layer (PML) is a technique to truncate the computational domain when
simulating acoustic wave propagating in an unbounded domain, which is firstly introduced
by Berenger [28]. When PML is introduced, the 3D acoustic wave equation (4.1) is modified
to
1
ρc2(utt + αut + βu+ γw)−∇ ·
(1
ρ(~v +∇u)
)= f,
~vt +H~v + J∇u−K∇w = 0,
wt − u = 0,
(4.63)
73
Figure 4.2: Snapshot of y-section of the acoustic pressure u at t = 0.9s in Example 4.3.2
where α = σx + σy + σz, β = σxσy + σxσz + σyσz, γ = σxσyσz and
H =
σx
σy
σz
, K =
σyσz
σxσz
σxσy
.
J =
σx − σy − σz
σy − σx − σz
σz − σx − σy
.
The detailed derivation of the wave equation with PML is included in appendix C.1. Here
σλ is a damping function with σι = 0 in the non-absorbing domain and σι 6= 0 in the
absorbing layer for ι = x, y, z. Also σι varies along ι axis only. The common choices for
the damping functions are constant functions, linear functions, quadratic functions, inverse
distance functions ([33, 35]).
In order to avoid the long time stability issue in 3D case [34], the 2D analogy of equation
(4.63) is solved to show the accuracy of the new finite difference scheme. The following
74
Figure 4.3: Snapshot of y-section of the acoustic pressure u at t = 1.4s in Example 4.3.2
substitution is used for second order temporal accuracy
u = e−12σtu := su. (4.64)
Then 2D analogy of equation (4.63) can be reformulated as
1
ρc2
(sutt +
(−1
4σ2 + ζ
)su
)−∇ ·
(1
ρ(~v +∇ (su))
)= f,
~vt +H~v + J∇(su) = 0.
(4.65)
The equation (4.65) is to be solved on the domain Ω = [0, 2π] × [0, 2π] and time t ∈ [0, 1]
with the following parameters
ρ = 1, c = 1, σx = sin(x)− 1, σy = sin(y)− 1. (4.66)
Note that ρ and c are chosen as constants so that the reference solution is available for error
calculation. It is worth to mention that the new scheme also works well for general cases
with variable ρ and c. The source term, the initial and boundary conditions are chosen
75
Figure 4.4: Snapshot of z-section of the acoustic pressure u at t = 0.4s in Example 4.3.2.
Table 4.2: Numerical errors in Example (4.3.3) in max norm. Here τ =(
5hπ
)2.
h π/25 π/50 π/75 π/100τ 1/25 1/100 1/225 1/400E 2.2419e-03 1.4182e-04 2.8029e-05 8.8773e-06
Conv.Order - 3.9826 3.9986 3.9966
accordingly to the analytic solution
u = et sin(x) sin(y). (4.67)
For simplicity, choose uniform spatial grid size hx = hy = hz = h and temporal step
size τ =(
5hπ
)2. The results presented in Table 4.2 clearly show that the new scheme is of
2nd-order in time and 4th-order in space with the truncation error O(τ 2) +O(h4).
4.3.4 Example 4
This example applies the new scheme to simulate the seismic wave propagation in the
Marmousi-2 model [39] using the acoustic wave equation with PML (4.63). The region
76
Figure 4.5: Snapshot of z-section of the acoustic pressure u at t = 0.9s in Example 4.3.2.
is a two dimensional domain with depth of 3500m and width of 17000m, which is discretized
using the spatial step hx = hy = 20m and time step size τ = 0.00125s. The velocity and
density are shown in Figure 4.7.
The PML is placed around the model with width 600m, as depicted in Figure 4.8.
Here the inverse distance damping function is given by
σx(x, z) =
0 in interior,
σmax
dPML−dist(x,z)+hxhx in PML.
(4.68)
where dist(x, z) represents the distance of point (x, z) to the interior domain. The thickness
of PML layer is dPML = 800m with σmax = 100 as a parameter. The seismic wave is generated
by a Ricker wavelet given by
s(t, x, z) = δ(x− x0, z − z0)(1− 2π2f 2
p (t− dr)2)e−π
2f2p (t−dr)2 , (4.69)
where the central frequency fp = 5Hz and time delay dr = 0.2s. The source is placed at the
77
Figure 4.6: Snapshot of z-section of the acoustic pressure u at t = 1.4s in Example 4.3.2
centre of the model x = 8500m and z = 1740m to show the absorbing effect of PML.
Four snapshots of the wavefields are shown in Figure 4.9 for t = 0.5s, 1.0s, 1.5s and 2.0s.
It can be shown in Figure 4.9(a) that the seismic wave has not arrived at the boundary when
t = 0.5s. Therefore, no wavefield can be observed except for a small region around the centre
of the domain. Then at t = 1.0s, one can see in Figure 4.9(b) that the wave approaches
the bottom of the domain and the wave energy has been absorbed by the perfectly matched
layer. Figure 4.9(c)(d) show that the seismic wave is traveling along the bottom boundary
to left and right without reflection due to PML.
To furthermore validate the effectiveness of the PML in absorbing energy for reflection
reduction, consider the acoustic energy of the acoustic wave
E(t) =
∫Ω
(ρ
2~v · ~v +
u2
2ρc2
)dxdz. (4.70)
Here c and ρ are the velocity and density in the Marmousi-2 model, and u is the pressure
which is computed when equation (4.63) is solved. The particle velocity ~v can be computed
78
Figure 4.7: Simulation of Marmousi-2 model. (a): acoustic velocity of the Marmousi-2model. (b): density of the Marmousi-2 model.
through the linear momentum conservation formula
ρ∂~v
∂t= −∇u. (4.71)
Applying Crank-Nicolson method to equation (4.71) leads to
~vn+1 = ~vn − τ
2ρ
∂xun + ∂xun+1
∂yun + ∂yu
n+1
. (4.72)
From the energy conservation of wave equation, the acoustic energy should be increasing at
first and then stay as a constant as time goes by if PML is not introduced into the model.
Figure 4.10 shows that the acoustic energy inside the domain is decreasing, which validates
that the PML indeed absorbs the energy as seismic wave encounters the boundary of the
domain.
79
Figure 4.8: Computation domain with PML zone: Ω = Ω0 ∪ ΩPML, where Ω0 is the originaldomain and ΩPML is the PML domain with width 800m
4.4 Conclusion
In this Chapter a new compact explicit finite difference has been developed to solve acoustic
wave equation with variable velocity speed and media density. The 2nd-order convergence in
time and 4th-order convergence in space have been theoretically analyzed and numerically
confirmed by several examples. The high efficiency of the numerical scheme is obtained
through the explicit treatment of the time derivative. The new scheme has a time complexity
which is linear to the total number of grid points for each time step. Because the spatial
differential operator is not self-adjoint, the widely used energy method is not applicable here
for stability analysis. To overcome this issue, we developed a stability analysis based on
empirical assumption and the spectrum estimation of the discrete differential operator in
space, to derive the Courant-Friedrichs-Lewy condition. This stability analysis shows that
the new scheme is conditionally stable. Furthermore, we applied the new scheme for the wave
equation with PML, which is a more realistic problem in Geophysics. Numerical results from
four examples clearly demonstrates that the new scheme is efficient, stable and accurate for
numerical simulation of seismic wave propagation, and is expected to find wide applications
in numerical seismic modeling and related areas.
80
Figure 4.9: Snapshots of the acoustic pressure u at different time.
81
Figure 4.10: Acoustic energy of the wavefield.
82
Chapter 5
Conclusion
This study investigated three compact finite difference schemes in series for acoustic wave
equations with variable acoustic velocity. All of these three schemes are of 4th-order in space
and conditionally stable.
The first scheme is implicit and used Pade approximation to obtain 4th-order accuracy.
Alternating Direction Implicit method is applied to solve the linear system at each time step
which greatly reduced the time complexity.
The second scheme employed so-called combined finite difference method to approximate
the second order derivatives. At each time step, tridiagonal linear systems have to be solved
but the overall time complexity is linear to the total number of grids. This scheme is much
simpler to implement than the first scheme.
The third scheme generalized the second scheme to make the combined finite difference
method also work for acoustic wave equations in variable density media, so that real world
models can also be solved by the new scheme.
For all the new schemes mentioned above, the stability conditions are obtained by either
energy method or analogy of von Neumann analysis. Numerical experiments are conducted
to validate the accuracy and efficiency of each scheme. Furthermore, the third scheme is also
applied to simulate a very realistic seismic model as an example.
83
The high-order accuracy of the new schemes can suppress numerical dispersion better
and require much less grid points per wavelength than conventional finite difference meth-
ods. Then less computational resources are costed when high-order schemes are applied
to solve wave equations, especially when solving large-scale 3D wave equations. Our new
schemes have the advantage of compactness when compared to many other high-order finite
difference methods. Thus only single layer of boundary conditions are needed. It can be
expected that the compact schemes will have even higher-order accuracy than non-compact
schemes if multiple layers of boundary conditions are imposed. Our another improvement is
the stability analysis which are usually missing for non-conventional finite difference schemes.
The conditional stability of the new schemes guarantees that no unexpected error accumula-
tion will occur when looking for long time numerical solution of wave equations. It is worth
to notice again that the third of the new schemes can solve wave equations with variable
coefficients not only in homogeneous media but also heterogeneous media, which is the most
common and most difficult case in real-world problems.
These new schemes are expected to find wide applications in numerical seismic modeling
and related areas, including both forward and inverse methods in seismic imaging, crude oil or
coal exploration, seabed exploration, etc. In particular, for Geophysics related Full Waveform
Inversion (FWI) where solving wave equations plays a critical role, our new schemes should
be able to greatly improve the solving speed and cost due to the above mentioned advantages
of the new schemes.
In the future, the authors will work on looking for better methods for the stability
analysis of the third scheme, so that the freezing coefficients assumption can be dropped and
the stability condition with PML boundary condition can also be obtained.
84
Appendix A
Appendix for Chapter 2
A.1 Proof of Theorem 2.2
Proof:
Firstly expand δ2t u
ni,j,k by Taylor series at time tn and the grid point (xi, yj, zk) as
δ2t u
ni,j,k = un+1
i,j,k − 2uni,j,k + un−1i,j,k
= τ 2 ∂2u
∂t2|ni,j,k +
τ 4
12
∂4u
∂t4|ni,j,k +
τ 6
360
∂6u(xi, yj, zk, τ∗n)
∂t6, (A.1)
where τ ∗n ∈ (tn−1, tn+1).
By Pade approximation, if v(x, y, z, t) is a sufficiently smooth function, we have
δ2y
1 + 112δ2y
vni,j,k = δ2y
(1−
δ2y
12
)vni,j,k +
h6y
144
∂6v(xi, y∗j , zk, tn)
∂y6, (A.2)
where y∗j ∈ (yj−1, yj+1).
85
On the other hand,
δ2y
(1− 1
12δ2y
)vni,j,k =
(δ2y −
1
12δ2yδ
2y
)vni,j,k
=−vni,j+2,k + 16vni,j+1,k − 30vni,j,k + 16vni,j−1,k − vni,j−2,k
12
= h2y
∂2v
∂y2|ni,j,k −
2
15h6y
∂6v(xi, y∗∗j , zk, tn)
∂y6, (A.3)
where y∗∗j ∈ (yj−1, yj+1).
Letting vni,j,k = δ2t u
ni,j,k and combining Eq. (A.4) with Eq. (A.3) lead to
δ2y
1 + 112δ2y
δ2t u
ni,j,k = h2
y
∂2v
∂y2|ni,j,k −
2h6y
15
∂6v(xi, y∗∗j , zk, tn)
∂y6+
h6y
144
∂6v(xi, y∗j , zk, tn)
∂y6. (A.4)
Using Eq. (A.1), it then follows that
λy ci,j,kδ2y
1 + 112δ2y
δ2t u
ni,j,k
= λy ci,j,kδ2y
1 + 112δ2y
[τ 2 ∂
2u
∂t2|ni,j,k +
τ 4
12
∂4u
∂t4|ni,j,k +
τ 6
360
∂6u(xi, yj, zk, τ∗n)
∂t6
]= τ 4
[ci,j,k
∂4u
∂y2∂t2|ni,j,k
]+O(h6
y) +O(τ 6) +O(τ 4h4y). (A.5)
Further, let w(x, y, z, t) = c(x, y, z) ∂4u∂y2∂t2
, then the first term on the right-hand side of
Eq. (2.22) can be written as
λx144
ci,j,kδ2x
1 + δ2x12
λy ci,j,kδ2y
1 +δ2y12
δ2t u
ni,j,k
= τ 4 ci,j,k144
[τ 2
h2x
δ2x
1 + 112δ2x
wni,j,k
]+O(h6
y) +O(τ 6) +O(τ 4h4y) (A.6)
= τ 4 ci,j,k144
[τ 2
h2x
(δ2x
(1− 1
12δ2x
)wni,j,k +O(h6
x)
)]+O(h6
y) +O(τ 6) +O(τ 4h4y).
86
Expanding δ2x(1− 1
12δ2x) w
ni,j,k we obtain
δ2x
(1− 1
12δ2x
)wni,j,k =
1
12
[−wni−2,j,k + 16wni−1,j,k − 30wni,j,k + 16wni+1,j,k − wni+2,j,k
]. (A.7)
Using Taylor series expansion, we can simplify it to
δ2x(1−
1
12δ2x) w
ni,j,k = h2
x
∂2w
∂x2|ni,j,k −
2h6x
15
∂6w(x∗i , yj, zk, tn)
∂x6, (A.8)
where x∗i ∈ (xi−1, xi+1).
Inserting Eq. (A.8) into Eq. (A.6) leads to
λx144
ci,j,kδ2x
1 + δ2x12
λy ci,j,kδ2y
1 +δ2y12
δ2t u
ni,j,k
= τ 4 ci,j,k144
[τ 2 ∂
2w
∂x2|ni,j,k
]+O(τ 6) +O(h6
y) +O(τ 6h4y)
= τ 6 ci,j,k144
∂2w
∂x2|ni,j,k +O(τ 6) +O(h6
y), (A.9)
where
∂2w
∂x2=∂2c
∂x2
∂4u
∂y2∂t2+ 2
∂c
∂x
∂5u
∂x∂y2∂t2+ c(x, y, z)
∂6u
∂x2∂y2∂t2. (A.10)
Similarly, we can derive the error estimations of other terms in Eq. (2.22) as the follows:
λy144
ci,j,kδ2y
1 +δ2y12
λz ci,j,kδ2z
1 + δ2z12
δ2t u
ni,j,k
= τ 4 ci,j,k144
[τ 2 ∂
2w
∂y2|ni,j,k
]+O(τ 6) +O(h6
z)
= τ 6 ci,j,k144
∂2w
∂y2|ni,j,k +O(τ 6) +O(h6
z), (A.11)
where
∂2w
∂y2=∂2c
∂y2
∂4u
∂z2∂t2+ 2
∂c
∂y
∂5u
∂y∂z2∂t2+ c(x, y, z)
∂6u
∂y2∂z2∂t2, (A.12)
87
λx144
ci,j,kδ2x
1 + δ2x12
λz ci,j,kδ2z
1 + δ2z12
δ2t u
ni,j,k
= τ 4 ci,j,k144
[τ 2 ∂
2w
∂x2|ni,j,k
]+O(τ 6) +O(h6
z)
= τ 6 ci,j,k144
∂2w
∂x2|ni,j,k +O(τ 6) +O(h6
z), (A.13)
where
∂2w
∂x2=∂2c
∂x2
∂4u
∂z2∂t2+ 2
∂c
∂x
∂5u
∂x∂z2∂t2+ c(x, y, z)
∂6u
∂x2∂z2∂t2, (A.14)
and
− λx1728
ci,j,kδ2x
1 + δ2x12
λy ci,j,kδ2y
1 +δ2y12
λz ci,j,kδ2z
1 + δ2z12
δ2t u
ni,j,k
= − τ 8
1728ci,j,k
[∂2
∂x2
(∂2w
∂y2
)]ni,j,k
+O(τ 6) +O(h6y), (A.15)
where ∂2w∂y2
is defined in Eq. (A.12). Therefore, the factoring error ERR in Eq. (2.22) is
given by
ERR = Mt τ6 + Mx h
6x + +My h
6y + +Mz h
6z, (A.16)
provided that the following functions
c(x, y, z),∂c(x, y, z)
∂xm1∂ym2∂zm3,
∂2c(x, y, z)
∂xn1∂yn2∂zn3
are bounded in Ω, where the non-negative integers satisfying m1 +m2 +m3 = 1 and n1 +n2 +
n3 = 2. Moreover, the solution u(x, y, z, t) and its derivatives ∂6u(x,y,z,t)
∂xk1∂yk2∂zk3∂tk4are bounded in
Ω × [0, T ], where the non-negative integers satisfy k1 + k2 + k3 + k4 = 6. Mt, Mx, My and
Mz are positive constants depending on the functions listed above.
88
Appendix B
Appendix for Chapter 3
B.1 Details of the Scheme
The numerical scheme for equation (3.3) in Section 3.1 is given by
un+1i,j,k = τ 2[ν2
i,j,k(∆u)ni,j,k + sni,j,k] + 2uni,j,k − un−1i,j,k (B.1)
for n > 0 and 1 6 i 6 Nx, 1 6 j 6 Ny, 1 6 k 6 Nz, with
u−1i,j,k =αi,j,k − τβi,j,k +
τ 2
2[ν2i,j,k(∆α)i,j,k + s0
i,j,k]
−τ3
6[ν2i,j,k(∆β)i,j,k + (∂ts)
0i,j,k] +O(τ 4)
(B.2)
and
(uxx)n∗,j,k = A−1
x
[1
h2x
(Bxu
n∗,j,k + b1q
b,nx
)− a1q
a,nx
](B.3)
(uyy)ni,∗,k = A−1
y
[1
h2y
(Byu
ni,∗,k + b1q
b,ny
)− a1q
a,ny
](B.4)
(uzz)ni,j,∗ = A−1
z
[1
h2z
(Bzu
ni,j,∗ + b1q
b,nz
)− a1q
a,nz
](B.5)
89
where
Aλ =
a0 a1
a1 a0 a1
. . . . . .
a1 a0 a1
a1 a0
Nλ×Nλ
, Bλ =
b0 b1
b1 b0 b1
. . . . . .
b1 b0 b1
b1 b0
Nλ×Nλ
(B.6)
for λ ∈ x, y, z and
un∗,j,k =
un1,j,k
un2,j,k...
unNx,j,k
Nx×1
, (uxx)n∗,j,k =
(uxx)n1,j,k
(uxx)n2,j,k
...
(uxx)nNx,j,k
Nx×1
, (B.7)
uni,∗,k =
uni,1,k
uni,2,k...
uni,Ny ,k
Ny×1
, (uyy)ni,∗,k =
(uyy)ni,1,k
(uyy)ni,2,k
...
(uyy)ni,Ny ,k
Ny×1
, (B.8)
uni,j,∗ =
uni,j,1
uni,j,2...
uni,j,Nz
Nz×1
, (uzz)ni,j,∗ =
(uzz)ni,j,1
(uzz)ni,j,2
...
(uzz)ni,j,Nz
Nz×1
. (B.9)
90
The q-vectors for boundary values in (B.3)(B.4)(B.5) are given by
qa,nx =
(uxx)n0,j,k
0
...
0
(uxx)nNx+1,j,k
Nx×1
, qb,nx =
(f0)nj,k
0
...
0
(f1)nj,k
Nx×1
, (B.10)
qa,ny =
(uyy)ni,0,k
0
...
0
(uyy)ni,Ny+1,k
Ny×1
, qb,ny =
(g0)ni,k
0
...
0
(g1)ni,k
Ny×1
, (B.11)
qa,nz =
(uzz)ni,j,0
0
...
0
(uzz)ni,j,Nz+1
Nz×1
, qb,nx =
(h0)ni,j
0
...
0
(h1)ni,j
Nz×1
. (B.12)
The boundary values for uxx, uyy and uzz above are given by
(uxx)n0,j,k =
1
ν20,j,k
[(∂2t f0)nj,k − sn0,j,k
]− (∂2
yf0)nj,k − (∂2zf0)nj,k (B.13)
(uxx)nNx+1,j,k =
1
ν2Nx+1,j,k
[(∂2t f1)nj,k − snNx+1,j,k
]− (∂2
yf1)nj,k − (∂2zf1)nj,k (B.14)
(uyy)ni,0,k =
1
ν2i,0,k
[(∂2t g0)ni,k − sni,0,k
]− (∂2
xg0)ni,k − (∂2zg0)ni,k (B.15)
(uyy)ni,Ny+1,k =
1
ν2i,Ny+1,k
[(∂2t g1)ni,k − sni,Ny+1,k
]− (∂2
xg1)ni,k − (∂2zg1)ni,k (B.16)
91
(uzz)ni,j,0 =
1
ν2i,j,0
[(∂2t h0)ni,j − sni,j,0
]− (∂2
xh0)ni,j − (∂2yh0)ni,j (B.17)
(uzz)ni,j,Nz+1 =
1
ν2i,j,Nz+1
[(∂2t h1)ni,j − sni,j,Nz+1
]− (∂2
xh1)ni,j − (∂2yh1)ni,j. (B.18)
B.2 Example on Boundary Values in Runge-Kutta Meth-
ods
This section briefly shows how to deal with the boundary values in implementing Runge-
Kutta Methods by an example. To ensure 4th-order spatial accuracy, all of the Lapla-
cian ∆ appears below should be considered as approximation by the same method as
(3.23)(3.24)(3.25). Thus some boundary values of K’s are necessary.
Consider the following RK4
Un+1 = Un + τ6
(K1 + 1
2K2 + 1
2K3 +K4
)K1 = AUn + S(tn)
K2 = A(Un + τ
2K1
)+ S(tn + 1
2τ)
K3 = A(Un + τ
2K2
)+ S(tn + 1
2τ)
K4 = A (Un + τK3) + S(tn + τ)
(B.19)
where
U =
u
ut
, A =
0 1
ν2∆ 0
, S =
0
s
, Kl =
Kl,1
Kl,2
. (B.20)
One will have
K11 =ut|tn
K12 =ν2∆u|tn + s|tn(B.21)
92
K21 =ut|tn +τ
2K12
K22 =ν2∆u|tn +τ
2ν2∆K11 + s|tn+ τ
2
(B.22)
K31 =ut|tn +τ
2K22
K32 =ν2∆u|tn +τ
2ν2∆K21 + s|tn+ τ
2
(B.23)
K41 =ut|tn + τK32
K42 =ν2∆u|tn + τν2∆K31 + s|tn+τ
(B.24)
Thus the boundary values of K11, K21, K31 are required to approximate ∆K11, ∆K21, ∆K31,
respectively. Check the expressions of K21 and K31, one may find that the boundary values
of K12 and K22 are also necessary.
By the equation (3.3), one has
K11|xmin =(∂tf0)|tn
K12|xmin =ν2∆u|tnxmin + s|tnxmin
=utt|tnxmin = (∂2t f0)|tn
(B.25)
Then K21|xmin will be easy to obtained. Note that (ν2∆K11)|xmin is necessary to evaluate
K22|xmin
(ν2∆K11)|xmin =(ν2∆ut|tn)|xmin =
[∂t(ν
2∆u)]|tn|xmin
=
[∂t(utt − s)] |tn
|xmin
=(∂3t f0 − s|xmin)|tn
(B.26)
The above equations used the fact that
ut|tnxmin =(∂tf0)|tn
ν2∆u|tnxmin =utt|tnxmin − s|tnxmin
= (∂2t f0)|tn − s|tnxmin .
(B.27)
93
Finally, K31|xmin will be easy to obtained as well. The boundary conditions at other bound-
aries can be obtained similarly.
In order to approximate ∆K31, ∆K21 and ∆K11, the boundary values of the second deriva-
tives of K31, K21 and K11 are necessary. Here is an example on how to obtain K31,xx|xmin ,
the other boundary values can be obtained similarly (Note that those boundary values for
the second derivatives of K31 are the most complicated cases. The K21 and K11 cases are
simpler).
From the expression of K31,xx, one can see that the following boundary values are neces-
sary (if not specified, the following expressions are all evaluated at t = tn)
ut,xx|xmin , K22,xx|xmin , (B.28)
Then from the expression of K22,xx, one needs
[∂2x(ν
2∆u)]|xmin ,
[∂2x(ν
2∆K11)]|xmin , sxx|tn+τ/2
xmin. (B.29)
From the expression of ν2∆K11, one needs
[∂2x(ν
2∆ut)]|xmin . (B.30)
The s-related boundary values are already known. For the u-related boundary values in
(B.28,B.29,B.30), one has (recall that uxx|xmin is given by (B.13))
ut,xx|xmin = ∂t(B.13)[∂2x(ν
2∆u)]|xmin =
[∂2x(utt − s)
]|xmin = ∂2
t (B.13)− sxx|xmin[∂2x(ν
2∆ut)]|xmin = ∂3
t (B.13)− stxx|xmin
. (B.31)
Thus K31,xx|xmin is obtained.
94
Appendix C
Appendix for Chapter 4
C.1 PML Derivation
The complex-valued coordinate stretching strategy [30] has been applied to derive the equa-
tion (4.63). Similar derivation can be found in [28]. To introduce the PML to the 3D acoustic
wave equation (4.1), replace the spatial differential operator ∂x, ∂y and ∂z by,
∂x −→1
1 + σxiω
∂x =1
ηx∂x,
∂y −→1
1 + σyiω
∂y =1
ηy∂y,
∂z −→1
1 + σziω
∂z =1
ηz∂z.
(C.1)
Here ω is the frequency. The damping function σx = 0 in the interior domain and σx 6= 0 in
the absorbing layer, and it does not change along y and z directions. The same properties
95
hold for σy and σz. Then the equation (4.1) becomes
ηxηyηz1
ρc2utt
−[(∂x
1
ρ
)(ηyηzηx
∂xu
)+
(∂y
1
ρ
)(ηxηzηy
∂yu
)+
(∂z
1
ρ
)(ηxηyηz
∂zu
)]−[
1
ρ∂x
(ηyηzηx
∂xu
)+
1
ρ∂y
(ηxηzηy
∂yu
)+
1
ρ∂z
(ηxηyηz
∂zu
)]= ηxηyηzs,
(C.2)
Here the support of s should be inside the non-absorbing domain, which means ηxηyηz = 1.
The temporal derivative terms in equation (C.2) is
ηxηyηzρc2
utt =1
ρc2
(1 +
σxiω
)(1 +
σyiω
)(1 +
σziω
)utt
=1
ρc2
(utt + (σx + σy + σz)ut + (σxσy + σxσz + σyσz)u+ σxσyσx
u
iω
),
(C.3)
One of the spatial derivative terms in equation (C.2) is
ηyηzηx
∂xu =
(1 + σy
iω
) (1 + σz
iω
)(1 + σx
iω
) ∂xu
=−σx + σy + σz + σyσz
iω
iω + σx∂xu+ ∂xu
:= vx + ∂xu,
(C.4)
and the other spatial derivatives are similar. For the term vx above, one has:
iωvx + σxvx + (σx − σy − σz)∂xu− σyσz∂( u
iω
)= 0. (C.5)
By Fourier transform, one sets iωvx = (vx)t. Then let w = uiω
, one has wt = u. Together
with equations (C.2), (C.3), (C.4), (C.5), and replacing the source term ηxηyηzs by s, the
equation system (4.63) can be derived.
96
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