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HC Chen 2/3/2020
Chapter 3A: Finite-Difference 1
Chapter 3
Preliminary Computational
Technique
Numerical MethodsAnalytic solution – linear equation, simple
geometry, simple initial and boundary conditionsAnalytic solution techniques – separation of
variables, Green function, Laplace transform, Theory of characteristics, …
Complex geometryComplex equations (nonlinear, coupled)Complex initial / boundary conditions
No analytic solutionsNumerical methods needed !!
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 2
Numerical MethodsObjective: Accuracy at minimum cost
(not at any cost!)
Numerical Accuracy (error analysis)Numerical Stability (stability analysis)Numerical Efficiency (minimize cost)Validation (model/prototype data, field data,
analytic solution, theory, asymptotic solution)Reliability and Flexibility (reduce preparation
and debugging time)Flow Visualization (graphics and animations)
OVERVIEWOverview of the computational solution procedures
Governing Equations ICS/BCS
DiscretizationSystem of Algebraic Equations
Equation (Matrix) Solver
Approximate Solution
Continuous Solutions
Finite-Difference
Finite-Volume
Finite-Element
Spectral
Boundary Element
Discrete Nodal Values
Tridiagonal
ADI
SOR
Gauss-Seidel
Conjugate gradient
Gaussian elimination
Ui (x,y,z,t)
p (x,y,z,t)
T (x,y,z,t)
or
(,,, )
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 3
3.1 Discretization1. Time derivativesalmost exclusively by finite-difference methods
2. Spatial derivativesFDM (Finite-Difference Methods)FVM (Finite-Volume Methods)FEM (Finite-Element Methods)FAM (Finite-Analytic Methods)Spectral MethodsBoundary Element Methods, etc.
Analytic solution
3.1.1 Converting Derivatives to Discrete Algebraic Equations
Heat Equation
Unsteady, one-dimensionalParabolic PDEMarching in time, elliptic in spaceThe simplest system to illustrate both the
“propagation” and “equilibrium” behaviors
1x0 xT0xT
dt1T bt0T
x
T
t
T
o
2
2
),(),(
),(,),(
T
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 4
Parabolic EquationConvective Transport Equation
T = temperature, = T = concentration, = DT = vorticity, = T = momentum, = T = turbulent kinetic energy, = + t
T = turbulent energy dissipation, = + t/
2
2
2
2
TTu
T
x
T
x
Tu
t
T
8
Fourier Analysis One-Dimensional Heat Equation
General solution in Fourier series
Characteristic polynomial
t x xxT uT T 0
jn x j t n2j n
1T x t T i x i t
4ˆ( , ) exp ( ) exp ( )
( ) ( )2 2t x x x t x
2x x
tt x
i iu i i u 0
0 (second derivative)0 dt
1dx (first derivative)u 0
u
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 5
9
Fourier Analysis Two-Dimensional Heat Equation
General solution in Fourier series
Characteristic polynomial
Pure diffusion (u=v=0)
t x y xx yyT uT vT T T 0( )
ˆ exp ( ) exp ( ) exp ( )jkn x j y k t n3j k n
1T T i x i y i t
8
2 2t x y x y
2 2x y t x y
2 2x y
t x y
i iu iv i i
i u v 0
0
u v 0
( ) ( )
( )
Elliptic in space
Characteristic surface
Advection-Diffusion Convective Transport Equation
2
2
x
T
x
Tu
t
T
Diffusion of pollutenin a still lake (u = 0)
Diffusion/convection of polluten in a river
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Chapter 3A: Finite-Difference 6
DiscretizationChoose suitable step size and time incrementReplace continuous information by discrete
nodal valuesConstruct discretization (algebraic) equations
with suitable numerical methodsSpecify appropriate auxiliary conditions for
discretization equationsClassification of PDE is importantSolve the system of “well-posed” equations by
matrix solver
)t,x(TT
)t,x(T
njnj :Numerical
:Exact
3.1.2 Spatial DerivativesFinite-difference: Taylor-series expansionFinite-element: low-order shape function
and interpolation function, continuous within each element
Finite-volume: integral form of PDE in each control volume
Spectral method: higher-order interpolation, continuous over the entire domain
Spectral element: finite-element/spectralPanel method, Boundary element method
Convert PDE to algebraic equations
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 7
3.1.3 Time DerivativesOne-sided (forward or backward) differences
Two-level schemesThree-level schemesRunge-Kutta mehtodsAdams-Bashforth-Moulton predictor-
corrector methodsUsually no advantages in using higher-order
integration formula unless the spatial discretization error can be improved to the same order
t
TT
t
T
t
TT
t
T1n
jnj
nj
1nj
or
Finite-Difference MethodsReplace derivatives by differences
j j+1 j+2j-1j-2
1j 2jj1j
2j
1jx 2jx jx1jx
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 8
2 31 2 3
21 2 3 1
2 3 2
( ) ( ) ( ) ( ) ( )
( ) 2 ( ) 3 ( ) ( )
( ) 2 6 ( ) ( ) / 2!
o o o o o o
o o o
o o
f x a a x x a x x a x x a f x
f x a a x x a x x a f x
f x a a x x a f x
3 3
( ) ( )1
0
( ) 6 ( ) / 3!
( ) ( !) ( 1) ( 1) 2 ( ) ( ) / !
( ) ( )
o
m mm m o m o
m om
f x a a f x
f x m a m m m a x x a f x m
f x a x x
( )
0
( )( )
!
mm mo
om
f xx x
m
Taylor series expansionConstruction of finite-difference formulaNumerical accuracy: discretization error
xo x
Truncation ErrorsTaylor series
Truncation error
How to reduce truncation errors?(a) Reduce grid spacing, use smaller x = xxo
(b) Increase order of accuracy, use larger n
( )
( ) ( )( ) ( ) ( ) ( ) ( ) ( )
! !
( )( ) ,
!
2 3o o
o o o o o
nno
o o
x x x xf x f x x x f x f x f x
2 3
x xf x a x b a x b
n
( )( )( ) ,
( )!
n 1n 1o
E
x xT f a b
n 1
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 9
Finite-Differences xo = xj , x = xj+1 = xj + x
xo x
( ) ( )( ) ( )
!
( ) ( )( ) ( )
!
mmo
om 0
m mj
j 1 j mm 0
f xf x x x
m
f x xf x f x x
x m
!
)()(
j0mm
mm
1jx
T
m
xxT
3.2 Approximation to Derivatives
Partial differential equations: dx, dtFinite-difference equations: x, tTime and spatial derivatives
(i) Taylor series expansion(ii) General Technique – Methods of
undetermined coefficientsDiscretization errors Numerical accuracy
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Chapter 3A: Finite-Difference 10
3.2.1 Taylor series expansionTruncated Taylor series – truncation error
x x
t
1njT
njT n
1jT n
1jT
j j+1j-1n
n+1
( ) )
!
( )
!
nm mMn 1 M 1
j 1mm 0 j
nm mMn M 1
j 1 2mm 0 j
t TT κ ( t
m t
x TT κ ( x)
m x
Truncation errors
Finite-Differences
Forward difference
Backward difference
Central difference
( ) ( ) ( )
! ! !
( )
! ! !
n n nnm m 2 2 3 3n n 4
j 1 jm 2 3m 0 jj j j
n nnm m 2 2 3 3n n
j 1 jm 2 3m 0 jj j
x T T x T x TT T x O x 1
m x x 2 x 3 x
x T T x T x TT T x
m x x 2 x 3 x
( ) ( ) n
4
j
O x 2
( )
( )
( )
( )
n n nj 1 j
j
n n nj j 1
j
n n nj 1 j 1 2
j
n n n n2j 1 j j 1 2
2 2
j
T TTO x
x x
T TTO x
x x
T TTO x
x 2 x
T 2T TTO x
x x
1
2
12
1+2
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 11
3.2.2 General TechniqueMethod of undetermined coefficients
General expression for discretization formula
3-point symmetric
3-point asymmetric
3-point asymmetric
4-point asymmetric
5-point symmetric
j2 j1 j j+1 j+2
)(
)(
)(
)(
)(
mn2j
n1j
nj
n1j
n2j
n
j
mn2j
n1j
nj
n1j
n
j
mn2j
n1j
nj
n
j
mn2j
n1j
nj
n
j
mn1j
nj
n1j
n
j
xOTeTdTcTbTax
T
xOTdTcTbTax
T
xOTcTbTax
T
xOTcTbTax
T
xOTcTbTax
T
2j1jj1j xxxx
General ProceduresExpand the functional values at xj-2, xj-1, xj+1,
xj+2, etc. about point xj (uniform spacing)
Can be generalized for non-uniform grids
n
j4
44n
j3
33
n
j2
22n
j
nj
n2j
n
j4
44n
j3
33
n
j2
22n
j
nj
n1j
x
T
4
x2
x
T
3
x2
x
T
2
x2
x
Tx2TT
x
T
4
x
x
T
3
x
x
T
2
x
x
TxTT
! !
!
! !
!
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 12
General ProceduresMethod of undetermined coefficients
Uniform or Non-uniform grid spacing
j4
44j
j3
33j
j2
22j
jjj1j
j4
441j
j3
331j
j2
221j
j1jj1j
x4
x
x3
x
x2
x
xx
x4
x
x3
x
x2
x
xx
! !
!
! !
!
Method of Undetermined Coefficients
2
3
4
!
!
!
j 1 j j 1 j
j j 1j
22 2j j 1
j
33 3j j 1
j
44 4j j 1
j
a b c a b c
a x c xx
1a x c x
2 x
1a x c x
3 x
1a x c x
4 x
j
1j
xaboutand
Expand
1j
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 13
3-point Symmetric Formula for First Derivative
For consistency
Leading term of truncation error
)Δ(ΔΔ
Δ
)Δ(ΔΔΔ
ΔΔ
)Δ(ΔΔ
Δ
1jj1j
j
1jj1jj
2j
21j
1jjj
1j
21j
2j
1jj
xxx
xc
xxxx
xxb
xxx
xa
0xcxa
1xcxa
0cba
33 ! x6
xx
x3
1xcxae
31jj
33
1j3jj
j
31jj
1jj1jj
j2j
21j1j
21j1j
2j
j x6
xx
xxxx
xxxx
x
3
3-point Asymmetric Formula
j
44
2j1j4
1j
j
33
2j1j3
1j
j
22
2j1j2
1j
j2j1j1j
j2j1jj
x4
1xxcxb
x3
1xxcxb
x2
1xxcxb
xxxcxb
cbacba
4
3
2
!
!
!
j2j1j xaboutandExpand
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 14
3-point Asymmetric Formula for First Derivative
For consistency
Leading term of truncation error
)(
)(
)(2a
2j1j2j
1j
2j1j
2j1j
2j1j1j
2j1j
22j1j
21j
2j1j1j
xxx
xc
xx
xxb
xxx
xx
0xxcxb
1xxcxb
0cba
33
)(
!
x6
xxx
x3
1xxcxbe
32j1j1j
33
2j1j3
1jj
j
32j1jj
2j1j2j1j
2j2
1j1j2
2j1jj2j1j2j
j
x6
xxx
xxxx
xxxxx2x
x
3
)(
Taylor Series Expansion
j6
66
j5
55
j4
44
j3
33
j2
22
jj2j
j6
66
j5
55
j4
44
j3
33
j2
22
jj1j
x6
x2
x5
x2
x4
x2
x3
x2
x2
x2
xx2
x6
x
x5
x
x4
x
x3
x
x2
x
xx
!
!
!
!
!
!
!
!
!
!
Uniform grid spacing
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 15
5-point Symmetric Formula
j
66
j
55
j
44
j
33
j
22
jj
2j1jj1j2j
x6
xe64dba64
x5
xe32dba32
x4
xe16dba16
x3
xe8dba8
x2
xe4dba4
xxe2dba2edcba
edcba
6
54
32
!
! !
! !
5-point Symmetric Formula for First Derivative
x3
2db
0cx12
1ea
0e16dba160e8dba8
0e4dba4xe2dba2
0edcba
1/
0e64dba64x
4e32dba32Leading term of
truncation error
Hx30
x88
x12
1
x 5
54
2j1j1j2jj
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 16
5-point Symmetric Formula for Second Derivative
2
2
2
x2
5c
x3
4db
x12
1ea
0e16dba160e8dba8
xe4dba4
0e2dba20edcba
2/ 2
2x
8e64dba64
0e32dba32Leading term of truncation error
Hx90
x163016
x12
1
x 6
64
2j1jj1j2j2j
2
2
3-point Symmetric Formula for First Derivative
x2
1db
0c
0dbxdb
0dcb 1/
0dbx
1dbLeading term of
truncation error
Hx6
x
x2
1
x 3
32
1j1jj
a = e = 0
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 17
3-point Symmetric Formula for Second Derivative
2
22
1b c d 0 b dxb d 0
2cb d 2/ x
x
2x
2db
0dbLeading term of truncation error
Hx12
x2
x
1
x 4
42
1jj1j2j
2
2
a = e = 0
3-point Asymmetric Formula for First Derivative
xx
xc
0e4dxe2d
0edc
Δ 21/e /2d
Δ 23/ 1/
x
2e8d
Leading term of truncation error
Hx3
x43
x2
1
x 3
32
2j1jjj
a = b = 0
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 18
3-point Asymmetric Formula for Second Derivative
2
2
2
2 x / 1e
x /2d
x /1c
x/2e4d
0e2d0edc
x
6e8d
Leading term of
truncation error
Hx
x2x
1
x 3
3
2j1jj2j
2
2
a = b = 0
3.3 Accuracy of the Discretization Process
Numerical Accuracy - truncation error leading term (Tables 3.3 and 3.4)
Example:
Truncation error
nn
n
2Ax2xx
Axx
Ax
xxxxxx
Ax
AeAAAe
uAe
1uTTu
;
Re ,Re
or
Tn
xA
x
T
n
xET
n
jn
nn
!
)(
!
)(..
Ax = Re u x = Rccell Reynolds number or cell Peclet number
L
x
U
uLUxuR
o
oc
****
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 19
xxx
xx
x eTeTeT , ,
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 20
Truncation ErrorsSlope: asymptotic rate-of-convergence
xxx
xx
x
eT
eT
eT
)log(loglog )( xmaExaE m
3.3.1 Higher-Order vs Low-Order Scheme
Higher-order formulaeHigher accuracy (for the same grid spacing)
Less efficient (for the same number of elements)
Less stable, may produce oscillatory solutions
May not be more accurate for discontinuity or severe gradients
Relatively little improvements for coarse-grid (large x) or lower-order (small m) solutions
)(
)(
!)(
)!()(
.).(
.).()(
)(
)(
)(
2m
11m
m
2m
1m
11m
m
1m
f
f
1m
x
m
xf
1m
xf
ET
ET
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 21
Numerical AccuracyCell Reynolds (or Peclet) number
Consider exponential function as an example
Require small Cell Reynolds number (Ax) or large m to achieve higher accuracy
Increasing m is not very effective in comparison with reducing Ax
1m
xA
TA
TA
1m
x
xT
xT
1m
x
ET
ET
TAeAx
TxT
eT
m
1m
m
1m
m
1m
mAxm
m
mm
Ax
)(
)(
.).(
.).(
)(
)(
)(
)(
Numerical Accuracy
x12
7
x12
1800
x12
TT8T8T
dx
Td :point-5
x2
1
x2
10
x2
TT
dx
Td :point-3
dx
Td :exact
2j1j1j2j
ax
1j1j
ax
ax
Higher-order formula is only
marginally better
x x x x
1
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 22
Numerical EfficiencyOverall efficiencyFast turnaround time
Limitations in computer memory (I/O) and execution time
It may not be feasible to use fine-grid all the time
Lower-order formula together with finer grid may be more effective than higher-order formula with coarse grid
Use non-uniform, adaptive grids
)](tanh[ x1ky
Use Non-uniform adaptive grid for efficient resolution of sharp gradients
HC Chen 2/3/2020
Chapter 3A: Finite-Difference 23
First-derivative Second-derivative
)](tanh[ x1ky