Substructuring domain decomposition scheme for unsteady problems

TL;DR

This paper introduces unconditionally stable, iteration-free domain decomposition schemes tailored for unsteady PDE problems, leveraging domain partitioning and two-component splitting to enhance parallel computation efficiency.

Contribution

It develops a novel class of unconditionally stable, iteration-free domain decomposition schemes based on domain partitioning and two-component splitting for unsteady problems.

Findings

Schemes are unconditionally stable for unsteady PDEs.

Numerical experiments demonstrate effectiveness on a 2D parabolic equation.

The approach improves parallel computation for time-dependent problems.

Abstract

Domain decomposition methods are used for approximate solving boundary problems for partial differential equations on parallel computing systems. Specific features of unsteady problems are taken into account in the most complete way in iteration-free schemes of domain decomposition. Regionally-additive schemes are based on different classes of splitting schemes. In this paper we highlight a class of domain decomposition schemes which is based on the partition of the initial domain into subdomains with common boundary nodes. Using the partition of unit we have constructed and studied unconditionally stable schemes of domain decomposition based on two-component splitting: the problem within subdomain and the problem at their boundaries. As an example there is considered the Cauchy problem for evolutionary equations of first and second order with non-negative self-adjoint operator in a finite Hilbert space. The theoretical consideration is supplemented with numerical solving a model problem for the two-dimensional parabolic equation.