# 28a 16 multi dimensional parabolic problemss i tl2 u

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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 Peaceman-Rachford 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 rst-order accurate as apparent from (5.2.9). Boundary conditions for the Douglas-Rachford 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 reaction-di 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 Multi-Dimensional 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. Springer-Verlag, Heidelberg, 1971. 19...
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## This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.

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