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Unformatted text preview: get the intermediate boundary condition. Let us illustrate this for the Dirichlet problem
(5.1.2) using the PeacemanRachford ADI scheme (5.2.6). During the horizontal sweep
(5.2.6a), we need boundary conditions for U +1 2 at x = 0 and 1. Adding (5.2.6a) and
(5.2.6b) gives
1
1
U +1 2 = 2 (I + 2t L2 )U + 2 (I ; 2t L2 )U +1 :
n = n n n n = = n n jk jk n jk Using the boundary condition (5.1.2c) U +1 2 = n = jk 1 (I + t L )
2
22 n + 1 (I ; 2t L2 ) +1
2
n jk jk (5.2.9) which can be used as a boundary condition for U +1 2 . The obvious boundary condition
n U +1 2 n jk = = = + +1
2
n n
jk jk is only rstorder accurate as apparent from (5.2.9).
Boundary conditions for the DouglasRachford scheme can be obtained from the corrector equation (5.2.8b) and the boundary condition (5.1.2c) as
^
U +1 n jk = (I ; tL2 ) +1 + tL2
n n jk jk : (5.2.10) Example 5.2.6. Strang 6] developed a factorization technique that has a faster rate
of convergence than the splitting (5.2.2) when the operators L1 and L2 do not commute. 5.2. Operator Splitting 17 Strang computes U +1 = e( 2)L2 e L1 e( n t= jk t t= 2)L2 U n
jk : (5.2.11) This scheme appears to require an extra solution per step however, if results are output
every n time steps then U = (e(
n t= jk 2)L2 e L1 e(
t or 2)L2 )(e( t= U = e( 2)L2 e L1 e( t= 2)L2 ) : : : (e( t t= 2)L2 e L1 e L2 e L1 : : : e( n t= jk t t t t= 2)L2 e L1 e( t= t 2)L2 U 0 jk t= 2)L2 )U 0 jk : Hence, the factorization (5.2.11) is the same as the simpler splitting (5.2.2) except for
the rst and last time steps.
Let us estimate the local error thus, assuming that U = u ,
n U +1 = (I + n jk U +1 n jk Thus, jk t + t2 2 + : : : )(I + t + t2 2 + : : : )
1
22 82
21
2
(I + t + t 2 + : : : )u
L L 2 L2 or n jk L 8 L2 L n jk 2
= I + t(L1 + L2) + 2t (L2 + L1L2 + L2L1 + L2) + O( t3 )]u :
1
2
n jk u +1 U
n jk ; +1 n jk = e (L1 +L2) ; e( 2)L2 e L1 e( 2)L1 ]u = O( t3):
t t= t t= n jk Problems
1. Although operator splitting has primarily been used for dimensional splitting, it
may be used in other ways. Consider the nonlinear reactiondi usion problem u = u + u(1 u)
t xx ; 0<x<1 u(0 t) = u(1 t) = 0
u(x 0) = (x) t>0 t>0
0 x 1: Develop a procedure for solving this problem that involves splitting the di usion
( u ) and reaction (u(1 ; u)) operators. Discuss its stability and local discretization errors.
xx 18 MultiDimensional Parabolic Problemss Bibliography
1] J. Douglas. On the numerical integration of
Joural of SIAM, 3:42{65, 1955. 2 @u
@ x2 + 2 @u
@ y2 = @u
@t by implicit methods. 2] J. Douglas and H.H. Rachford. On the numerical solution of heat conduction problems
in two and three space variables. Trasactions of the American Mathematics Society,
82:421{439, 1956.
3] E. Isaacson and H.B. Keller. Analysis of Numerical Methods. John Wiley and Sons,
New York, 1966.
4] A.R. Mitchell and D.F. Gri ths. The Finite Di erence Method in Partial Di erential
Equations. John Wiley and Sons, Chichester, 1980.
5] D.W. Peaceman and Jr. H.H. Rachford. The numerical solution of parabolic and
elliptic equations. Journal of SIAM, 3:28{41, 1955.
6] G. Strang. On the construction and comparison of di erence schemes. SIAM Journal
on Numerical Analysis, 5:506{517, 1968.
7] J.C. Strikwerda. Finite Di erence Schemes and Partial Di erential Equations. Chapman and Hall, Paci c Grove, 1989.
8] N. Yanenko. The Method of Fractional Steps. SpringerVerlag, Heidelberg, 1971. 19...
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 Spring '14
 JosephE.Flaherty

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