28a 16 multi dimensional parabolic problemss i tl2 u

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online