Year: 2019
Author: Frédéric Hecht, Sidi-Mahmoud Kaber
Journal of Mathematical Study, Vol. 52 (2019), Iss. 3 : pp. 244–257
Abstract
We are interested in the design of parallel numerical schemes for linear systems. We give an effective solution to this problem in the following case: the matrix $A$ of the linear system is the product of $p$ nonsingular matrices $A_i^m$ with specific shape: $A_i= I -{h_i}X$ for a fixed matrix $X$ and real numbers $h_i$. Although having a special form, these matrices $A_i$ arise frequently in the discretization of evolutionary Partial Differential Equations. For example, one step of the implicit Euler scheme for the evolution equation $u' =Xu$ reads $(I-hX)u^{n+1}=u^n$. Iterating $m$ times such a scheme leads to a linear system $Au^{n+m}=u^n$. The idea is to express $A^{-1}$ as a linear combination of elementary matrices $A_i^{-1}$ (or more generally in term of matrices $A_i^{-k}$). Hence the solution of the linear system with matrix $A$ is a linear combination of the solutions of linear systems with matrices $A_i$ (or $A_i^k$). These systems are then solved simultaneously on different processors.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jms.v52n3.19.02
Journal of Mathematical Study, Vol. 52 (2019), Iss. 3 : pp. 244–257
Published online: 2019-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 14
Keywords: Partial differential equations parabolic equation finite element methods finite difference methods parallel computing.