Finite diﬀerence method
Transcript of Finite diﬀerence method

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Finite difference method
Principle: derivatives in the partial differential equation are approximatedby linear combinations of function values at the grid points
1D: = (0 , X ), u i u (x i ), i = 0 , 1, . . . , N
grid points x i = i x mesh size x = XN
Firstorder derivatives
ux (x ) = lim x 0
u (x + x ) u (x) x = lim x 0
u (x ) u (x x) x
= lim x 0
u (x + x ) u (x x)2 x
(by denition)

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Approximation of rstorder derivativesGeometric interpretation
ux i
u i +1 u i x forward difference
ux i
u i u i 1 x backward difference
ux i
u i +1 u i 12 x central difference
Taylor series expansion u (x ) =
n =0
(x x i ) n
n ! n ux n i , u C
([0, X ])
T 1 : u i+1 = u i + xux i +
( x )2
2 2ux 2 i +
( x )3
6 3ux 3 i + . . .
T 2 : u i 1 = u i xux i
+( x)2
2 2ux 2 i
( x)3
6 3ux 3 i
+ . . .

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Analysis of truncation errorsAccuracy of nite difference approximations
T 1 u
x i=
u i+1 u i
x
x
2
2u
x2
i
( x)2
6
3u
x3
i+ . . .
forward difference truncation error O( x)
T 2 ux i
=u i u i 1
x+
x2
2ux 2 i
( x )2
6 3ux 3 i
+ . . .
backward difference truncation error O( x)
T 1 T 2 ux i
=u i+1 u i 1
2 x
( x )2
6 3ux 3 i
+ . . .
central difference truncation error O( x )2
Leading truncation error
= m ( x)m + m +1 ( x)m +1 + . . . m ( x )m

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Approximation of secondorder derivatives
Central difference scheme
T 1 + T 2 2ux 2 i
=u i+1 2u i + u i 1
( x)2+ O( x )2
Alternative derivation
2ux 2 i
=
xux i
= lim x 0
ux i+1 / 2
ux i 1/ 2
x
u i +1 u i
x u i u i 1
x x
= u i+1 2u i + u i 1
( x )2
Variable coefficients f (x ) = d(x ) ux diffusive ux
f x i
f i+1 / 2 f i 1/ 2
x= di+1 / 2
u i +1 u i x di 1/ 2
u i u i 1 x
x
=di+1 / 2u i+1 (di+1 / 2 + di 1/ 2)u i + di 1/ 2u i 1
( x)2

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Approximation of mixed derivatives
2D: 2 u
xy =
xuy =
y
ux
2 u
xy i,j = uy i +1 ,j uy i 1 ,j2 x + O ( x )2
uy i +1 ,j =u i +1 ,j +1 u i +1 ,j 1
2 y+ O ( y)2
uy i 1 ,j =u i 1 ,j +1 u i 1 ,j 1
2 y+ O ( y)2
Secondorder difference approximation
2uxy i,j
=u i+1 ,j +1 u i+1 ,j 1 u i 1,j +1 + u i 1,j 1
4 x y+ O[( x)2 , ( y)2]

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Onesided nite differences
ux 0
=u 1 u0
x+ O( x ) forward difference
backward/central difference approximationswould need u 1 which is not available
Polynomial tting
u (x) = u 0 + xu
x 0+
x2
2
2u
x2
0
+x3
6
3u
x3
0
+ . . .
u (x ) a + bx + cx 2 ,ux
b + 2 cx,ux 0
b
approximate u by a polynomial and differentiate it to obtain the derivatives
u 0 = a
u 1 = a + b x + c x 2
u 2 = a + 2 b x + 4 c x2
c x 2 = u 1 u 0 b x
b = 3u 0 + 4 u 1 u 2
2 x

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Analysis of the truncation error
Onesided approximationux i
u i + u i+1 + u i+2
x
u i+1 = u i + xux i
+( x )2
2 2ux 2 i
+( x)3
6 3ux 3 i
+ . . .
u i+2 = u i + 2 xu
x i +(2 x)2
2
2u
x 2 i +(2 x )3
6
3u
x 3 i + . . .
u i + u i +1 + u i +2 x =
+ + x u i + ( + 2 )
ux i +
x2 ( + 4 )
2 ux 2 i
+ O( x 2)
Secondorder accurate if + + = 0 , + 2 = 1 , + 4 = 0
= 32 , = 2 , = 12
ux i =
3u i +4 u i +1 u i +22 x + O( x
2)

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Application to secondorder derivatives
Onesided approximation 2ux 2 i
u i + u i+1 + u i+2
x2
u i+1 = u i + xux i
+( x )2
2 2ux 2 i
+( x)3
6 3ux 3 i
+ . . .
u i+2 = u i + 2 xu
x i +(2 x)2
2
2u
x 2 i +(2 x )3
6
3u
x 3 i + . . .
u i + u i +1 + u i +2 x 2 =
+ + x 2 u i +
+2 x
ux i +
+4 2
2 ux 2 i
+ O( x)
Firstorder accurate if + + = 0 , + 2 = 0 , + 4 = 2
= 1 , = 2, = 1 ux i =u i 2u i +1 + u i +2
x 2 + O( x )

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Highorder approximations
ux i
=2u i+1 + 3 u i 6u i 1 + u i 2
6 x+ O( x)3 backward difference
ux i
= u i+2 + 6 u i+1 3u i 2u i 16 x
+ O( x)3 forward difference
ux i
= u i+2 + 8 u i+1 8u i 1 + u i 2
12 x+ O( x )4 central difference
2ux 2 i
= u i+2 + 16 u i+1 30u i + 16 u i 1 u i 2
12( x)2+ O( x )4 central difference
Pros and cons of highorder difference schemes
more grid points, llin, considerable overhead cost high resolution, reasonable accuracy on coarse grids
Criterion: total computational cost to achieve a prescribed accuracy

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Example: 1D Poisson equation
Boundary value problem
2u
x 2= f in = (0 , 1), u(0) = u(1) = 0
Onedimensional mesh
u i u (x i ), f i = f (x i ) x i = i x, x = 1N , i = 0 , 1, . . . , N
Central difference approximation O( x )2
u i 1 2u i + u i +1( x ) 2 = f i , i = 1 , . . . , N 1
u 0 = u N = 0 Dirichlet boundary conditions
Result: the original PDE is replaced by a linear system for nodal values

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Example: 1D Poisson equation
Linear system for the central difference scheme
i = 1 u 0 2u 1 + u 2( x ) 2 = f 1
i = 2 u 1 2u 2 + u 3
( x ) 2 = f 2i = 3 u 2 2u 3 + u 4( x ) 2 = f 3
. . .
i = N 1 u N 2 2u N 1 + u N( x ) 2 = f N 1
Matrix form Au = F A R N 1 N 1 u, F R N 1
A =1
( x )2
2 1 1 2 1
1 2 1. . . 1 2
, u =
u 1u 2u 3
u N 1
, F =
f 1f 2f 3
f N 1
The matrix A is tridiagonal and symmetric positive denite invertible.

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Other types of boundary conditionsDirichletNeumann BC u (0) = ux (1) = 0
u 0 = 0 ,u N +1 u N 1
2 x= 0 u N +1 = u N 1 central difference
Extra equation for the last node
u N 1 2u N + u N +1
( x)2= f N
uN 1 + u N ( x)2
=12
f N
Extended linear system Au = F A R N N u, F R N
A =1
( x)2
2 1 1 2 1
1 2 1. . . 1 2 1
1 1
, u =
u1u2u3
uN 1u N
, F =
f 1f 2f 3
f N 112 f N
The matrix A remains tridiagonal and symmetric positive denite.

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Other types of boundary conditions
Nonhomogeneous Dirichlet BC u (0) = g0 only F changes
u0 = g0 2u1 u 2
( x)2= f 1 +
g0( x )2
rst equation
Nonhomogeneous Neumann BC ux (1) = g1 only F changes
uN +1 u N 12 x
= g1 u N +1 = u N 1 + 2 xg 1
u N 1 2u N + u N +1
( x)2= f N u
N 1 + u N ( x)2
= 12
f N + g1
x
Nonhomogeneous Robin BC ux (1) + u (1) = g2 A and F change
u N +1 u N
12 x + u N = g2 u N +1 = u N
1 2 xu N + 2 xg 2
u N 1 2u N + u N +1
( x )2= f N
u N 1 + (1 + x )uN ( x )2
=12
f N +g2 x

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Example: 2D Poisson equation
Boundary value problem
2 u
x 2 2 uy 2 = f in = (0 , 1) (0, 1)
u = 0 on =
Uniform mesh: x = y = h, N = 1h
u i,j u(x i , y j ), f i,j = f (x i , y j ), (x i , y j ) = ( ih,jh ), i, j = 0 , 1, . . . , N
Central difference approximation O(h 2)
u i 1 ,j + u i,j 1 4u i,j + u i +1 ,j + u i,j +1h 2 = f i,j , i, j = 1 , . . . , N 1
u i, 0 = u i,N = u 0,j = uN,j = 0 i, j = 0 , 1, . . . , N

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Example: 2D Poisson equation
Linear system Au = F A R (N 1)2 (N 1) 2 u, F R (N 1)
2
rowbyrownode numbering
u = [u 1,1 . . . u N 1,1 u 1,2 . . . u N 1,2 u1,3 . . . u N 1,N 1]T
F = [ f 1,1 . . . f N 1,1 f 1,2 . . . f N 1,2 f 1,3 . . . f N 1,N 1]T
A =
B I I B I
. . . . . .
I B I I B
, B =
4 1 1 4 1
. . . . . .
1 4 1 1 4
I =
11
11
The matrix A is sparse, blocktridiagonal(for the above numbering) and SPD.
cond2(A) = max  min 
= O(h 2)
Caution: convergence of iterative solvers deteriorates as the mesh is rened

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Treatment of complex geometries2D Poisson equation
2 u
x 2 2 uy 2 = f in
u = g0 on
Difference equation
u 1 + u2 4u0 + u 3 + u 4
h 2= f 0
curvilinear boundary
Q
P
Q04
h
2
1
3
R
stencil of Q
Linear interpolation
u (R ) =u 4(h ) + u 0
h= g0(R ) u 4 = u 0
h
+ g0(R )h
h
Substitution yields u 1 + u 2 4 + h u 0 + u3 = h2 f 0 + g0(R ) hh
Neumann and Robin BC are even more difficult to implement

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Grid transformations
Purpose: to provide a simple treatment of curvilinear boundaries
The original PDE must be rewritten in terms of ( , ) instead of ( x, y ) anddiscretized in the computational domain rather than the physical one.
Derivative transformationsux
,uy
, . . .
difficult to compute
u
,u
, . . .
easy to compute

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PDE transformations for a direct mapping
Direct mapping = (x, y ), = (x, y )
Chain rule ux
=u
x
+u
x
,uy
=u
y
+u
y
2ux 2
=u
2x 2
+u
2x 2
+ 2 2u
x
x
+ 2u 2
x
2+
2u 2
x
2
2uy 2
=u
2y 2
+u
2y 2
+ 2 2u
y
y
+ 2u 2
y
2
+ 2u 2
y
2
Example: 2D Poisson equation u = f turns into
2u 2
x
2
+y
2
2u 2
x
2
+y
2
2 2u
x
x
+y
y
u
2x 2
+ 2y 2
u
2x 2
+ 2y 2
= f transformed equationscontain many more terms
The metrics need to be determined (approximated by nite differences)

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Direct versus inverse mappingTotal differentials for both coordinate systems
= (x, y )
= (x, y )
d = x dx +y dy
d = x dx + y dy
d
d=
x
y
x y
dx
dy
x = x (, )
y = y(, )
dx = x d +x d
dy = y
d + y
d
dx
dy=
x
x
y
y
d
d
x
y
x
y
=x
x
y
y
1
=1
det J
y
x
yx
Relationship between the direct and inverse metricsx
=1
det J y
,x
= 1
det J y
,y
= 1
det J x
,y
=1
det J x