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Discontinuous Galerkin Methods forHamilton-Jacobi Equations
Chi-Wang Shu
Division of Applied Mathematics
Brown University
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Outline
• Introduction of discontinuous Galerkin method for conservation laws
• Discontinuous Galerkin method for Hamilton-Jacobi equations of Huand Shu, reinterpreted by Li and Shu
• Discontinuous Galerkin method for Hamilton-Jacobi equations ofCheng and Shu
• Fast sweeping method for steady state solutions
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Introduction of discontinuous Galerkin method
How does the method work — an example
To solve a hyperbolic conservation law:
ut + f(u)x = 0 (1)
Multiplying with a test function v, integrate over a cell Ij = [xj− 12
, xj+ 12
],
and integrate by parts:∫
Ij
utvdx−∫
Ij
f(u)vxdx+ f(uj+ 12
)vj+ 12
− f(uj− 12
)vj− 12
= 0
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Now assume both the solution u and the test function v come from a finite
dimensional approximation space Vh, which is usually taken as the space
of piecewise polynomials of degree up to k:
Vh ={
v : v|Ij ∈ P k(Ij), j = 1, · · · , N}
However, the boundary terms f(uj+ 12
), vj+ 12
etc. are not well defined
when u and v are in this space, as they are discontinuous at the cell
interfaces.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
From the conservation and stability (upwinding) considerations, we take
• A single valued monotone numerical flux to replace f(uj+ 12
):
f̂j+ 12
= f̂(u−j+ 1
2
, u+j+ 1
2
)
where f̂(u, u) = f(u) (consistency); f̂(↑, ↓) (monotonicity) and f̂ isLipschitz continuous with respect to both arguments.
• Values from inside Ij for the test function v
v−j+ 1
2
, v+j− 1
2
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Hence the DG scheme is: find u ∈ Vh such that∫
Ij
utvdx−∫
Ij
f(u)vxdx+ f̂j+ 12
v−j+ 1
2
− f̂j− 12
v+j− 1
2
= 0 (2)
for all v ∈ Vh.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Time discretization could be by the TVD Runge-Kutta method (Shu and
Osher, JCP 1988). For the semi-discrete scheme:
du
dt= L(u)
where L(u) is a discretization of the spatial operator, the third order TVD
Runge-Kutta is simply:
u(1) = un + ∆tL(un)
u(2) =3
4un +
1
4u(1) +
1
4∆tL(u(1))
un+1 =1
3un +
2
3u(2) +
2
3∆tL(u(2))
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Advantages of the DG method:
• Easy handling of complicated geometry and boundary conditions(common to all finite element methods). Allowing hanging nodes in the
mesh;
• Compact. Communication only with immediate neighbors, regardlessof the order of the scheme;
• Explicit. Because of the discontinuous basis, the mass matrix is localto the cell, resulting in explicit time stepping (no systems to solve);
• Parallel efficiency. Achieves 99% parallel efficiency for static mesh andover 80% parallel efficiency for dynamic load balancing with adaptive
meshes (Flaherty et al.);
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
• Provable cell entropy inequality and L2 stability, for arbitrary scalarequations in any spatial dimension and any triangulation, for any order
of accuracy, without limiters;
• At least (k + 12)-th order accurate, and often (k + 1)-th order accurate
for smooth solutions when piecewise polynomials of degree k are
used, regardless of the structure of the meshes.
• Easy h-p adaptivity.
• Stable and convergent DG methods are now available for manynonlinear PDEs containing higher derivatives: convection diffusion
equations, KdV equations, ...
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Three examples
We show three examples to demonstrate the excellent performance of the
DG method.
The first example is the linear convection equation
ut + ux = 0, or ut + ux + uy = 0,
on the domain (0, 2π) × (0, T ) or (0, 2π)2 × (0, T ) with thecharacteristic function of the interval (π
2, 3π
2) or the square (π
2, 3π
2)2 as
initial condition and periodic boundary conditions.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
0 1 2 3 4 5 6x
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1u
k=1, t=100π, solid line: exact solution;dashed line / squares: numerical solution
0 1 2 3 4 5 6x
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1u
k=6, t=100π, solid line: exact solution;dashed line / squares: numerical solution
Figure 1: Transport equation: Comparison of the exact and the RKDG so-
lutions at T = 100π with second order (P 1, left) and seventh order (P 6,
right) RKDG methods. One dimensional results with 40 cells, exact solution
(solid line) and numerical solution (dashed line and symbols, one point per
cell)
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
0
0.2
0.4
0.6
0.8
1
1.2
u
0
1
2
3
4
5
6
x
0
1
2
3
4
5
6
y
P1
0
0.2
0.4
0.6
0.8
1
1.2
u
0
1
2
3
4
5
6
x
0
1
2
3
4
5
6
y
P6
Figure 2: Transport equation: Comparison of the exact and the RKDG so-
lutions at T = 100π with second order (P 1, left) and seventh order (P 6,
right) RKDG methods. Two dimensional results with 40 × 40 cells.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
The second example is the double Mach reflection problem for the two
dimensional compressible Euler equations.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
Rectangles P1, ∆ x = ∆ y = 1/240
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
Rectangles P2, ∆ x = ∆ y = 1/240
Figure 3: Double Mach reflection. ∆x = ∆y = 1240
. Top: P 1; bottom:
P 2.Division of Applied Mathematics, Brown University
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
2.0 2.2 2.4 2.6 2.8
0.0
0.1
0.2
0.3
0.4
0.5
Rectangles P2, ∆ x = ∆ y = 1/240
2.0 2.2 2.4 2.6 2.8
0.0
0.1
0.2
0.3
0.4
Rectangles P1, ∆ x = ∆ y = 1/480
Figure 4: Double Mach reflection. Zoomed-in region. Top: P 2 with ∆x =
∆y = 1240
; bottom: P 1 with ∆x = ∆y = 1480
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
2.0 2.2 2.4 2.6 2.8
0.0
0.1
0.2
0.3
0.4
0.5
Rectangles P2, ∆ x = ∆ y = 1/240
2.0 2.2 2.4 2.6 2.8
0.0
0.1
0.2
0.3
0.4
Rectangles P2, ∆ x = ∆ y = 1/480Rectangles P2, ∆ x = ∆ y = 1/480
Figure 5: Double Mach reflection. Zoomed-in region. P 2 elements. Top:
∆x = ∆y = 1240
; bottom: ∆x = ∆y = 1480
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
The third example is the flow past a forward-facing step problem for the
two dimensional compressible Euler equations. No special treatment is
performed near the corner singularity.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
Rectangles P1, ∆ x = ∆ y = 1/320
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
Rectangles P2, ∆ x = ∆ y = 1/320
Figure 6: Forward facing step. Zoomed-in region. ∆x = ∆y = 1320
. Left:
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
History of the DG method:
Here is a (very incomplete) history of the study of DG methods:
• 1973: First discontinuous Galerkin method for steady state linearscalar conservation laws (Reed and Hill).
• 1974: First error estimate (for tensor product mesh) of thediscontinuous Galerkin method of Reed and Hill (LeSaint and Raviart).
• 1986: Error estimates for discontinuous Galerkin method of Reed andHill (Johnson and Pitkäranta).
• 1989-1998: Runge-Kutta discontinuous Galerkin method for nonlinearconservation laws (Cockburn, Shu, ...).
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
• 1994: Proof of cell entropy inequality for discontinuous Galerkinmethod for nonlinear conservation laws in general multidimensional
triangulations (Jiang and Shu).
• 1997-1998: Discontinuous Galerkin method for convection diffusionproblems (Bassi and Rebay, Cockburn and Shu, Baumann and Oden,
...).
• 2002: Discontinuous Galerkin method for partial differential equationswith third or higher order spatial derivatives (KdV, biharmonic, ...) (Yan
and Shu, Xu and Shu, ...)
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Collected works on the DG methods:
• Discontinuous Galerkin Methods: Theory, Computation andApplications, B. Cockburn, G. Karniadakis and C.-W. Shu, editors,
Lecture Notes in Computational Science and Engineering, volume 11,
Springer, 2000. (Proceedings of the first DG Conference)
• Journal of Scientific Computing, special issue on DG methods, 2005.
• Computer Methods in Applied Mechanics and Engineering, specialissue on DG methods, 2006.
• Journal of Scientific Computing, special issue on DG methods, toappear in 2009.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
DG method for Hamilton-Jacobi equations of Hu and Shu (SISC 1999)
Reinterpreted by Li and Shu (Appl. Math. Let 2005)
Idea: convert HJ to conservation laws and then use DG
In one space dimension the HJ equation is
φt +H(φx) = 0, φ(x, 0) = φ0(x). (3)
This is a relatively easy case because (3) is equivalent to the conservation
law
ut +H(u)x = 0, u(x, 0) = u0(x) (4)
if we identify u = φx.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Assuming (3) is solved in the interval a ≤ x ≤ b and it is divided into thefollowing cells:
a = x 12
< x 32
< · · · < xN+ 12
= b, (5)
we denote
Ij = (xj− 12
, xj+ 12
), xj =1
2
(
xj− 12
+ xj+ 12
)
, (6)
hj = xj+ 12
− xj− 12
, h = maxjhj ,
and define the following approximation space
V kh ={
v : v|Ij ∈ P k(Ij), j = 1, ..., N}
. (7)
Here P k(Ij) is the set of all polynomials of degree at most k on the cell
Ij .
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
A k-th order discontinuous Galerkin scheme for the one dimensional
Hamilton-Jacobi equation (3) can then be defined as follows: find ϕ ∈ V kh ,such that∫
Ij
ϕxtv dx−∫
Ij
H(ϕx)vx dx+Ĥj+ 12
v−j+ 1
2
−Ĥj− 12
v+j− 1
2
= 0, j = 1, ..., N,
(8)
for all v ∈ V k−1h . Here
Ĥj+ 12
= Ĥ(
(ϕx)−
j+ 12
, (ϕx)+j+ 1
2
)
(9)
is a monotone flux, Ĥ(↑, ↓). Notice that the method described above isexactly the discontinuous Galerkin method for the conservation law
equation (4) satisfied by the derivative u = φx.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
The DG scheme described above only determines ϕ for each element up
to a constant, since it is only a scheme for ϕx. The missing constant can
be obtained in one of the following two ways:
1. By requiring that∫
Ij
(ϕt +H(ϕx)) v dx = 0, j = 1, ..., N, (10)
for all v ∈ V 0h , that is,∫
Ij
(ϕt +H(ϕx)) dx = 0, j = 1, ..., N. (11)
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
2. By using (11) to update only one (or a few) elements, e.g., the
left-most element I1, then use
ϕ(xj, t) = ϕ(x1, t) +
∫ xj
x1
ϕx(x, t) dx (12)
to determine the missing constant for the cell Ij .
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Both approaches are used in our numerical experiments. They perform
similarly for smooth problems, with the first approach giving slightly better
results. However, it is our numerical experience that, when there are
singularities in the derivatives, the first approach will often produce dents
and bumps when the integral path in time passes through the singularities
at some earlier time. The philosophy of using the second approach is that
one could update only a few elements whose time integral paths do not
cross derivative singularities.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Lemma 2.1.. The following L2 stability result for the derivative ϕx holds
for the discontinuous Galerkin method defined above, of any order of
accuracy k applied to any nonlinear Hamilton-Jacobi equation (3):
d
dt
∫ b
a
ϕ2x dx ≤ 0. (13)
�
For a finite interval [a, b], (13) trivially implies TVB (total variation
bounded) property for the numerical solution ϕ:
TV (ϕ) =
∫ b
a
|ϕx| dx ≤√b− a
√
∫ b
a
(
d
dxφ0(x)
)2
dx. (14)
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
This is a rather strong stability result, considering that it applies even if the
derivative of the solution φx develops discontinuities, no limiter has been
added to the numerical scheme, and the scheme can be of arbitrary high
order in accuracy. It also implies convergence of at least a subsequence of
the numerical solution ϕ when h→ 0. However, this stability result is notstrong enough to imply that the limit solution is the viscosity solution of (3).
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Two Dimensional Case
We consider in this section the case of two spatial dimensions. The
algorithm in more spatial dimensions is similar. This time, the scalar
Hamilton-Jacobi equation
φt +H(φx, φy) = 0, φ(x, y, 0) = φ0(x, y) (15)
is in some sense equivalent to the following conservation law system
ut+H(u, v)x = 0, vt+H(u, v)y = 0, (u, v)(x, y, 0) = (u, v)0(x, y).
(16)
if we identify
(u, v) = (φx, φy). (17)
For example, a vanishing viscosity solution of (15) corresponds, via (17),
to a vanishing viscosity solution of (16), and vice versa.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
However, (16) is not a strictly hyperbolic system, which may cause
problems in its numerical solution if we treat u and v as independent
variables. Instead, we would like to still use φ as our solution variable (a
polynomial) and take its derivatives as u and v.
Assuming we are solving (15) in the domain Ω, which has a triangulation
Th consisting of triangles or general polygons of maximum size (diameter)h, with the following approximation space
V kh ={
v : v|K ∈ P k(K), ∀K ∈ Th}
, (18)
where P k(K) is again the set of all polynomials of degree at most k on
the cell K .
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
We propose a discontinuous Galerkin method for (15) as follows: find
ϕ ∈ V kh , such that∫
K
ϕxtv dxdy −∫
K
H(ϕx, ϕy)vx dxdy +∑
e∈∂K
∫
e
Ĥ1,e,K v dΓ = 0
(19)
and∫
K
ϕytv dxdy −∫
K
H(ϕx, ϕy)vy dxdy +∑
e∈∂K
∫
e
Ĥ2,e,K v dΓ = 0
(20)
for all v ∈ V k−1h and all K ∈ Th, in a least square sense. Here thenumerical flux is
Ĥi,e,K = Ĥi,e,K(
(∇ϕ)int(K), (∇ϕ)ext(K))
, i = 1, 2 (21)
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
where the superscript int(K) implies that the value is taken from within
the element K , and the superscript ext(K) implies that the value is taken
from outside the element K and within the neighboring element K ′
sharing the edge e with K . The flux (21) satisfies the following properties:
1. Ĥi,e,K is Lipschitz continuous with respect to all its arguments;
2. Consistency:
Ĥi,e,K (∇φ,∇φ) = H(∇φ)ni,where n = (n1, n2) is the unit outward normal to the edge e of the
element K ;
3. Conservation:
Ĥi,e,K(
(∇ϕ)int(K), (∇ϕ)ext(K))
= −Ĥi,e,K′(
(∇ϕ)int(K′), (∇ϕ)ext(K′))
.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
where K ∩K ′ = e.We will again mainly use the simple (local) Lax-Friedrichs flux
Ĥ1,e,K((u, v)−, (u, v)+) = H
(
u− + u+
2,v− + v+
2
)
n1−1
2α(u+−u−)
(22)
and
Ĥ2,e,K((u, v)−, (u, v)+) = H
(
u− + u+
2,v− + v+
2
)
n2−1
2β(v+−v−)
(23)
where α = maxu,v |∂H(u,v)∂u | and β = maxu,v |∂H(u,v)∂v
|, with themaximum being taken over the relevant (local) range.
Notice that (19)-(20) are exactly the discontinuous Galerkin method for the
conservation law system (16) satisfied by the derivatives (u, v) = ∇φ.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
For a rectangular mesh and for k = 1 this recovers the (local)
Lax-Friedrichs monotone scheme for (15) if we identify
uij =φi+1,j−φi−1,j
2∆xand vij =
φi,j+1−φi,j−12∆y
.
Of course, (19)-(20) have more equations than the number of degrees of
freedoms (an over-determined system) for k > 1, thus a least square
solution is needed. In practice, the least square procedure is performed as
follows: we first evolve (16) for one time step (one inner stage for high
order Runge-Kutta methods), using the discontinuous Galerkin method
(19)-(20) with u = ϕx and v = ϕy ; then ϕ at the next time level (stage) is
obtained (up to a constant) by least square:
||(ϕx−u)2+(ϕy−v)2||L1(K) = minψ∈Pk(K)
||(ψx−u)2+(ψy−v)2||L1(K).
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Important note: In Li and Shu (Appl. Math. Let. 2005), the DG space for
(u, v) is chosen to contain only locally curl-free polynomial pairs. This
yields the same algorithm mathematically as described above, however it
has the following two advantages:
• The least square procedure is no longer needed, as now the numberof the degree of freedom is equal to the number of equations
• The computational cost is reduced, because the locally curl-freespace is smaller than the original unconstrained space
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
This determines ϕ for each element up to a constant, since it is only a
scheme for ∇ϕ. The missing constant can again be obtained in one of thefollowing two ways:
1. By requiring that∫
K
(ϕt +H(ϕx, ϕy)) v dxdy = 0, (24)
for all v ∈ V 0h and for all K ∈ Th, that is,∫
K
(ϕt +H(ϕx, ϕy)) dxdy = 0, ∀K ∈ Th ; (25)
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
2. By using (25) to update only one (or a few) elements, e.g., the corner
element(s), then use
ϕ(B, t) = ϕ(A, t) +
∫ B
A
(ϕx dx+ ϕy dy) (26)
to determine the missing constant. The path should be taken to avoid
crossing a derivative discontinuity, if possible.
We remark that the procedure discussed above is easily implemented in
any triangulation, e.g., for both rectangles and triangles.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
DG method for Hamilton-Jacobi equations of Cheng and Shu (JCP 2007)
Hamilton-Jacobi equation:
ϕt +H(ϕx1, . . . , ϕxd , x1, . . . , xd) = 0, ϕ(x, 0) = ϕ0(x)
• We only consider linear or convex H as a function of ϕ in this lecture.
• Generic solution are continuous but may admit discontinuousderivatives.
• We are only interested in the viscosity solution, which is the uniquepractically relevant solution and satisfies the entropy condition.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Available numerical methods on structured meshes:
• Essentially non-oscillatory (ENO) or weighted ENO (WENO) finitedifference schemes (Osher and Shu SINUM 1991; Jiang and Peng
SISC 2000) on structured meshes:
– very simple
– highly accurate
– non-oscillatory
– very popular in applications
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
• WENO finite difference schemes on unstructured meshes (Zhang andShu SISC 2003):
– flexible in geometry and adaptive meshes
– accurate
– non-oscillatory
– rather complicated in coding
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
• DG schemes of Hu and Shu (SISC 1999)– First convert a HJ equation ϕt+H(ϕx) = 0 to a conservation law
ut +H(u)x = 0 (27)
by defining u = ϕx, then apply the usual DG scheme to (27) plus
evolving the cell average of ϕ.
– In multi-dimensions, a scalar HJ equation is converted to a system,
which is not strongly hyperbolic at u = 0. Additional cost and
complexity related to systems.
– Reinterpretation by Li and Shu (Appl. Math. Letters 2005): using
locally curl-free DG basis to save cost and to avoid the least square
procedure to reconstruct ϕ from u.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
A new DG scheme (Cheng and Shu, JCP 2007): works directly on the HJ
equation without first converting it to a hyperbolic system for the
derivatives.
Motivation: look at a simple linear HJ equation
ϕt + a(x)ϕx = 0
with a′(x) ≥ 0. It can be rewritten as a conservation law with a sourceterm
ϕt + (a(x)ϕ)x = a′(x)ϕ (28)
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
The usual DG method can be applied to (28) as finding ϕh ∈ Vh such that∫
Ij
∂tϕh(x, t)vh(x)dx−∫
Ij
a(x)ϕh(x, t)∂xvh(x)dx
+a(xj+ 12
)ϕh(x−
j+ 12
, t)vh(x−
j+ 12
) − a(xj− 12
)ϕh(x−
j− 12
, t)vh(x+j− 1
2
)
=
∫
Ij
a′(x)ϕh(x, t)vh(x)dx (29)
holds for all test functions vh(x) ∈ Vh. The scheme (29) can be rewrittenas
∫
Ij
(∂tϕh(x, t) + a(x)∂xϕh(x, t)) vh(x)dx
+a(xj− 12
)(
ϕh(x+j− 1
2
, t) − ϕh(x−j− 12
, t))
vh(x+j− 1
2
) = 0
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
This motivates the following definition of our DG scheme for general HJ
equations: Find ϕh ∈ Vh, such that∫
Ij
(∂tϕh(x, t) +H(∂xϕh(x, t), x))vh(x)dx
+1
2
(
minx∈I
j+12
H1(∂xϕh, xj+ 12
) −∣
∣
∣
∣
∣
minx∈I
j+12
H1(∂xϕh, xj+ 12
)
∣
∣
∣
∣
∣
)
[ϕh]j+ 12
(vh)−
j+ 12
+1
2
(
maxx∈I
j− 12
H1(∂xϕh, xj− 12
) +
∣
∣
∣
∣
∣
maxx∈I
j− 12
H1(∂xϕh, xj− 12
)
∣
∣
∣
∣
∣
)
[ϕh]j− 12
(vh)+j− 1
2
= 0, j = 1, . . . , N
holds for any vh ∈ Vh.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Remarks about the scheme:
• We need ∂xϕh on the cells Ij− 12
and Ij+ 12
, which include the points
xj− 12
and xj+ 12
respectively, where the numerical solution ϕh(x, t) is
discontinuous. We use a L2 type projection to reconstruct a
continuous approximation of ϕh(x, t).
• The purpose of taking the maximum and minimum is to obtain betterstability by adding more viscosity, while still maintaining accuracy.
• The scheme is Roe type which may generate entropy violatingsolutions. We perform an entropy correction in sonic expanding cells
by simply switching to the Hu-Shu DG scheme in those cells.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
We have the following stability and accuracy results for our scheme
applied to linear HJ equation
ϕt + a(x)ϕx = 0
Proposition 1. Suppose there is a constant β such that the derivative of
a(x) satisfies ax(x) < β for x ∈ [a, b], then we have the following L2stability for our scheme:
||ϕh(t)||L2 ≤ eβt/2||ϕh(0)||L2
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Proposition 2. If a(x) and the solution ϕ are smooth and the scheme
with the finite element space consisting of piecewise polynomials of
degree ≤ k is used, then we have the following optimal L2 error estimate
||ϕh(t) − ϕ(t)||L2 ≤ Chk+1
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Numerical results
Example 1. We solve the one dimensional problem
ϕt + sin(x)ϕx = 0
ϕ(x, 0) = sin(x)
ϕ(0, t) = ϕ(2π, t)
We clearly observe (k + 1)-th order of accuracy for P k polynomials.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Table 1: Errors and numerical orders of accuracy for Example 1 when us-
ing P 0 polynomials and Runge-Kutta third order time discretization on a
uniform mesh of N cells. Final time t = 1. CFL = 0.9.
N L1 error order L2 error order L∞ error order
40 0.49E-01 0.62E-01 0.29E+00
80 0.25E-01 0.95 0.32E-01 0.93 0.16E+00 0.86
160 0.13E-01 0.97 0.17E-01 0.96 0.83E-01 0.96
320 0.65E-02 0.98 0.84E-02 0.98 0.42E-01 0.99
640 0.33E-02 0.99 0.42E-02 0.99 0.21E-01 1.00
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Table 2: Errors and numerical orders of accuracy for Example 1 when us-
ing P 1 polynomials and Runge-Kutta third order time discretization on a
uniform mesh of N cells. Final time t = 1. CFL = 0.3.
N L1 error order L2 error order L∞ error order
40 0.12E-02 0.25E-02 0.15E-01
80 0.31E-03 1.96 0.68E-03 1.90 0.43E-02 1.81
160 0.78E-04 1.97 0.18E-03 1.94 0.11E-02 1.92
320 0.20E-04 1.98 0.46E-04 1.97 0.29E-03 1.96
640 0.50E-05 1.99 0.12E-04 1.98 0.74E-04 1.98
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Table 3: Errors and numerical orders of accuracy for Example 1 when us-
ing P 2 polynomials and Runge-Kutta third order time discretization on a
uniform mesh of N cells. Final time t = 1. CFL = 0.1.
N L1 error order L2 error order L∞ error order
40 0.48E-04 0.10E-03 0.52E-03
80 0.60E-05 2.99 0.14E-04 2.88 0.88E-04 2.58
160 0.75E-06 3.00 0.18E-05 2.90 0.14E-04 2.70
320 0.94E-07 2.99 0.24E-06 2.93 0.20E-05 2.78
640 0.12E-07 2.99 0.31E-07 2.95 0.27E-06 2.85
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Table 4: Errors and numerical orders of accuracy for Example 1 when us-
ing P 3 polynomials and Runge-Kutta third order time discretization on a
uniform mesh of N cells. Final time t = 1. CFL = 0.05.
N L1 error order L2 error order L∞ error order
40 0.21E-05 0.51E-05 0.29E-04
80 0.14E-06 3.96 0.35E-06 3.88 0.22E-05 3.75
160 0.87E-08 3.97 0.23E-07 3.93 0.16E-06 3.78
320 0.55E-09 3.97 0.15E-08 3.96 0.10E-07 3.91
640 0.35E-10 3.98 0.94E-10 3.98 0.68E-09 3.95
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Example 2.
We solve the two dimensional linear Hamilton-Jacobi equation with
variable coefficients
ϕt − yϕx + xϕy = 0.The computational domain is [−1, 1]2. The initial condition is given by
ϕ0(x, y) =
0 0.3 ≤ r0.3 − r 0.1 < r < 0.30.2 r ≤ 0.1
where r =√
(x− 0.4)2 + (y − 0.4)2. We also impose periodicboundary condition on the domain. This is a solid body rotation around the
origin.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
For this problem, the derivatives of ϕ are not continuous. Therefore, we do
not expect to obtain (k + 1)-th order of accuracy for P k polynomials, see
the following table for the result of P 2 elements.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Table 5: Errors and numerical orders of accuracy for Example 2 when us-
ing P 2 polynomials and Runge-Kutta third order time discretization on a
uniform mesh of N ×N cells. Final time t = 1. CFL = 0.1.
N ×N L1 error order L2 error order L∞ error order20 × 20 0.41E-03 0.13E-02 0.11E-0140 × 40 0.14E-03 1.58 0.55E-03 1.26 0.65E-02 0.8280 × 80 0.47E-04 1.54 0.24E-03 1.22 0.36E-02 0.84
160 × 160 0.15E-04 1.62 0.10E-03 1.23 0.21E-02 0.81
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
At t = 2π, i.e. the period of rotation, we take a snapshot at the line x = y
in Figure 7. We can see that a higher order scheme can yield better
results for this nonsmooth initial condition.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
x
ϕ
-0.5 0 0.5
0
0.05
0.1
0.15
0.2
xϕ
-0.5 0 0.5
0
0.05
0.1
0.15
0.2
Figure 7: Example 2. 80× 80 uniform mesh. t = 2π. Solid line: the exactsolution; Rectangles: the numerical solution. One dimensional cut of 45◦
with the x axis. Left: P 1 polynomial; Right: P 2 polynomial.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Example 3.
We solve the same two dimensional linear Hamilton-Jacobi equation with
variable coefficients
ϕt − yϕx + xϕy = 0as in the previous example but with a different initial condition as
ϕ0(x, y) = exp
(
−(x− 0.4)2 + (y − 0.4)22σ2
)
We take σ = 0.05 such that at the domain boundary, ϕ is very small,
hence imposing a periodic boundary condition will lead to small
non-smoothness errors. We then observe the desired order of accuracy in
the following table.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Table 6: Errors and numerical orders of accuracy for Example 3 when us-
ing P 2 polynomials and Runge-Kutta third order time discretization on a
uniform mesh of N ×N cells. Final time t = 1. CFL = 0.1.
N ×N L1 error order L2 error order L∞ error order20 × 20 0.14E-02 0.10E-01 0.28E+0040 × 40 0.15E-03 3.21 0.15E-02 2.81 0.53E-01 2.4180 × 80 0.11E-04 3.82 0.11E-03 3.73 0.58E-02 3.19
160 × 160 0.11E-05 3.30 0.12E-04 3.26 0.90E-03 2.69
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Example 4.
We solve the two dimensional linear Hamilton-Jacobi equation with
variable coefficients
ϕt + f(x, y, t)ϕx + g(x, y, t)ϕy = 0
with
f(x, y, t) = sin2(πx) sin(2πy) cos
(
t
Tπ
)
,
g(x, y, t) = − sin2(πy) sin(2πx) cos(
t
Tπ
)
where T is the period of deformation.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
The initial condition is given by
ϕ0(x, y) =
0 0.3 ≤ r0.3 − r 0.1 < r < 0.30.2 r ≤ 0.1
where r =√
(x− 0.4)2 + (y − 0.4)2. This is a numerical test forincompressible flow. During the evolution, the initial data is severely
deformed, then it returns to the original shape after one period. At
t = 1.5, i.e. the period of rotation, we take a snapshot at the line x = y in
Figure 8. We can clearly observe that a higher order scheme yields better
results for this nonsmooth initial condition.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
x
ϕ
-0.5 0 0.5
0
0.05
0.1
0.15
0.2
xϕ
-0.5 0 0.5
0
0.05
0.1
0.15
0.2
Figure 8: Example 4. 80×80 uniform mesh. t = 1.5. Solid line: the exactsolution; Rectangles: the numerical solution. One dimensional cut of 45◦
with the x axis. Left: P 1 polynomial; Right: P 2 polynomial.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Example 5.
We solve the model linear problem with a discontinuous coefficient
ϕt + sign(cos(x))ϕx = 0
ϕ(x, 0) = sin(x)
ϕ(0, t) = ϕ(2π, t)
For the viscosity solution, at x = π2
, there will be a shock forming in ϕx,
and at x = 3π2
, there is a rarefaction wave. For this example we need an
entropy correction.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
x
ϕ
1 2 3 4 5 6
-1
-0.5
0
0.5
xϕ
0 1 2 3 4 5 6
-1
-0.5
0
0.5
Figure 9: Example 5. t = 1, CFL = 0.1, N = 80, using P 2 polynomi-
als. Solid line: the exact solution; Rectangles: the numerical solution. Left:
without entropy correction; Right: with entropy correction.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Example 6.
One dimensional Burgers’ equation
ϕt +ϕ2x2
= 0
ϕ(x, 0) = sin(x)
ϕ(0, t) = ϕ(2π, t)
When t = 0.5, the solution is still smooth, and the expected third order
accuracy is obtained for P 2 polynomials, see Table 7. After t = 1, a
shock will form in ϕx, our scheme can resolve the derivative singularity
sharply, see Figure 10.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Table 7: Errors and numerical orders of accuracy for Example 6 when us-
ing P 2 polynomials and Runge-Kutta third order time discretization on a
uniform mesh of N cells. Final time t = 0.5. CFL = 0.1.
N L1 error order L2 error order L∞ error order
40 0.13E-04 0.22E-04 0.84E-04
80 0.17E-05 2.97 0.29E-05 2.93 0.12E-04 2.86
160 0.22E-06 2.98 0.37E-06 2.96 0.15E-05 2.92
320 0.27E-07 2.98 0.47E-07 2.97 0.20E-06 2.95
640 0.34E-08 2.99 0.59E-08 2.99 0.25E-07 2.97
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
x
ϕ
1 2 3 4 5 6-1
-0.5
0
0.5
Figure 10: Example 6. Numerical solution. Solid line: N = 500; Rectan-
gles: N = 40. Final time t = 1.5, CFL = 0.05, P 2 polynomials.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Example 7.
One dimensional Burgers’ equation with a nonsmooth initial condition
ϕt +ϕ2x2
= 0
ϕ(x, 0) =
π − x if 0 ≤ x ≤ πx− π elsewhere in[0, 2π],
ϕ(0, t) = ϕ(2π, t)
For the viscosity solution, the sharp corner at π will be smoothed out, and
a rarefaction wave will form in the derivative. For this example we also
need the entropy correction.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
x
ϕ
1 2 3 4 5 6
-0.5
0
0.5
1
1.5
2
2.5
xϕ
1 2 3 4 5 6
-0.5
0
0.5
1
1.5
2
2.5
Figure 11: Example 7. t = 1, CFL = 0.05, N = 80, using P 2 poly-
nomials. Solid line: the exact solution; Rectangles: the numerical solution.
Left: without entropy correction; Right: with entropy correction.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Example 8.
Two dimensional Burgers’ equation.
ϕt +(ϕx + ϕy + 1)
2
2= 0
ϕ(x, y, 0) = − cos(x+ y)
with periodic boundary condition on the domain [0, 2π]2.
We use a uniform rectangular mesh. At t = 0.1, the solution is still
smooth. Numerical errors and order of accuracy are listed in Table 8,
demonstrating the expected order of accuracy. At t = 1, the solution is no
longer smooth. We plot the numerical solution in Figure 12. We observe
good resolution of the kinks in the solution.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Table 8: Errors and numerical orders of accuracy for Example 8 when us-
ing P 2 polynomials and Runge-Kutta third order time discretization on a
uniform mesh of N ×N cells. Final time t = 0.1. CFL = 0.1.
N ×N L1 error order L2 error order L∞ error order10 × 10 0.30E-02 0.43E-02 0.35E-0120 × 20 0.38E-03 2.98 0.58E-03 2.90 0.56E-02 2.6440 × 40 0.48E-04 2.97 0.77E-04 2.91 0.80E-03 2.8180 × 80 0.66E-05 2.87 0.11E-04 2.83 0.14E-03 2.55
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
x
0
2
4
6
y
0
2
4
6
ϕ
-1.4
-1.2
-1
-0.8
-0.6
Figure 12: Example 8. Numerical solution when t = 1, CFL = 0.1,
40 × 40 uniform mesh, using P 2 polynomials.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Example 9.
We solve the Eikonal equation given by
ϕt + |ϕx| = 0ϕ(x, 0) = sin(x)
ϕ(0, t) = ϕ(2π, t)
Figure 13 shows the space-time location where the entropy correction is
applied. We observe that the correction is mostly applied at a few cells
neighboring the boundary of the rarefaction wave. The number of cells in
which the correction is performed is relatively small compared to the total
number of cells.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
x
t
1 2 3 4 5 60
2
4
6
8
Figure 13: Example 9. t = 10, CFL = 0.1, N = 160 uniform mesh,
using P 2 polynomials. ε = 10−10. Rectangular symbols mark the cells in
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Example 10.
We solve the two dimensional Eikonal equation
ϕt +√
ϕ2x + ϕ2y = 1
First, we consider the case of the computational domain being
[0, 1]2 \ [0.4, 0.6]2. For the inner boundary along [0.4, 0.6]2, we imposethe boundary condition ϕ = 0. On the other hand, we impose free outflow
boundary conditions on the outer boundary. The initial condition is taken
as ϕ0(x, y) = max{|x− 0.5|, |y − 0.5|} − 0.1. The steady statesolution should give us a function that is equal to the distance of the point
to the inner boundary. We plot the numerical steady state solution in
Figure 14.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
x
0
0.2
0.4
0.6
0.8
1
y
0
0.2
0.4
0.6
0.8
1
ϕ
0
0.1
0.2
0.3
0.4
0.5
x
y
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
Figure 14: Example 10. Steady state solution with 40 × 40 uniform mesh,using P 2 polynomials. Left: three dimensional plot; Right: contour plot.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Next, we consider this example with a point source condition; namely, we
take the inner boundary to be the center point (0.5, 0.5). In this case, we
would need ϕ in the center cell to be the L2 projection of the exact
distance function. For all other cells, the computation is the same as for
the previous case. The initial condition is taken as
ϕ0(x, y) = max{|x− 0.5|, |y − 0.5|}. We plot the numerical steadystate solution in Figure 15. We can see that in both cases we obtain very
good resolution to the viscosity solution.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
x
0
0.2
0.4
0.6
0.8
y
0
0.2
0.4
0.6
0.8
ϕ
0
0.2
0.4
0.6
x
y
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
Figure 15: Example 10. Steady state solution with 39 × 39 uniform mesh,using P 2 polynomials. Left: three dimensional plot; Right: contour plot.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
Fast sweeping method for steady state solutions
The DG method is quite local regardless of the order of accuracy, hence it
is well suited for the design of fast sweeping methods. Preliminary results
in Li, Shu, Zhang and Zhao (JCP 2008) for a P 1 fast sweeping DG
algorithm for the Eikonal equation based on the DG formulation of Cheng
and Shu are very promising. We have now obtained even better results,
with the number of iterations uniformly low (usually only 4 or at most 8
iterations) for all meshes.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
References for the third lecture:
[1] Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element
method for directly solving the Hamilton-Jacobi equations, Journal of
Computational Physics, v223 (2007), pp.398-415.
[2] C. Hu and C.-W. Shu, A discontinuous Galerkin finite element method
for Hamilton-Jacobi equations, SIAM Journal on Scientific Computing, v21
(1999), pp.666-690.
[3] O. Lepsky, C. Hu and C.-W. Shu, Analysis of the discontinuous Galerkin
method for Hamilton-Jacobi equations, Applied Numerical Mathematics,
v33 (2000), pp.423-434.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
[4] F. Li and C.-W. Shu, Reinterpretation and simplified implementation of a
discontinuous Galerkin method for Hamilton-Jacobi equations, Applied
Mathematics Letters, v18 (2005), pp.1204-1209.
[5] F. Li, C.-W. Shu, Y.-T. Zhang and H. Zhao, A second order
discontinuous Galerkin fast sweeping method for Eikonal equations,
Journal of Computational Physics, v227 (2008), pp.8191-8208.
[6] C.-W. Shu, High order numerical methods for time dependent
Hamilton-Jacobi equations, in Mathematics and Computation in Imaging
Science and Information Processing, S.S. Goh, A. Ron and Z. Shen,
Editors, Lecture Notes Series, Institute for Mathematical Sciences,
National University of Singapore, volume 11, World Scientific Press,
Singapore, 2007, pp.47-91.
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DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
The End
THANK YOU!
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