zbMATH — the first resource for mathematics

ADI method – domain decomposition. (English) Zbl 1099.65084
A domain decomposition algorithm which is based on an implicit prediction and fully implicit scheme for the interior values, for solving parabolic partial differential equations, is presented. It is shown that this algorithm without the correction procedure is unconditionally stable. The modified alternating direction implicit (ADI) algorithm is deduced from this algorithm so that the ADI method is considered as the extreme case of the domain decomposition algorithm.

MSC:
 65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35K15 Initial value problems for second-order parabolic equations
Full Text:
References:
 [1] Blum, H.; Lisky, S.; Rannacher, R., A domain splitting algorithm for parabolic problems, Computing, 49, 11-23, (1992) · Zbl 0767.65073 [2] Dawson, C.N.; Du, Q.; Dupont, T.F., A finite difference domain decomposition algorithm for numerical solution of the heat equation, Math. comp., 57, 195, 63-71, (1991) · Zbl 0732.65091 [3] Gustafsson, I.A., A class of first order factorization methods, Bit, 18, 142-156, (1978) · Zbl 0386.65006 [4] Jiang, H.; Wong, Y.S., A parallel alternating direction implicit preconditioning method, J. comput. appl. math., 36, 209-226, (1991) · Zbl 0734.65027 [5] Y. Jun, T.-Z. Mai, IPIC domain decomposition algorithm for parabolic problems, Preprint · Zbl 1094.65097 [6] Meijerink, J.A.; van der Vorst, H.A., An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. comp., 31, 148-162, (1977) · Zbl 0349.65020 [7] G.A. Meurant, Numerical experiments with a domain decomposition method for parabolic problems on parallel computers, in: Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, 1990, pp. 394-408 · Zbl 0766.65080 [8] Meurant, G.A., A domain decomposition method for parabolic problems, Appl. numer. math., 8, 427-441, (1991) · Zbl 0745.65056 [9] Peaceman, D.W.; Rachford, H.H., The numerical solution of parabolic and elliptic differential equations, J. soc. indust. appl. math., 8, 1, 28-41, (1955) · Zbl 0067.35801 [10] Smith, G.D., Numerical solution of partial differential equations: finite difference methods, (1985), Clarendon Press Oxford · Zbl 0576.65089 [11] Starke, G., Alternating direction preconditioning for nonsymmetric systems of linear equations, SIAM J. sci. comput., 15, 2, 369-384, (1994) · Zbl 0807.65024 [12] Thomas, J.W., Numerical partial differential equationsâ€”finite difference methods, (1995), Springer Berlin · Zbl 0831.65087 [13] Womble, D.E., A time-stepping algorithm for parallel computers, SIAM J. sci. statist. comput., 11, 5, 824-837, (1990) · Zbl 0705.65070 [14] Young, D.M., Iterative solution of large linear systems, (2003), Dover New York [15] Young, D.M.; Kincaid, D.R., A new class of parallel alternating-type iterative methods, J. comput. appl. math., 74, 331-344, (1996) · Zbl 0873.65020 [16] Young, D.M.; Mai, T.Z., Iterative algorithms and software for solving large sparse linear systems, Comm. appl. numer. methods, 4, 435-456, (1988) · Zbl 0644.65023 [17] J. Zhu, H. Qian, On an efficient parallel algorithm for solving time dependent partial differential equations, in: PDPTA ’98 International Conference, 1998, pp. 394-401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.