Polynomial collocation using a domain decomposition solution to parabolic PDE's via the penalty method and explicit/implicit time marching

Abstract

A domain decomposition method is examined to solve a time-dependent parabolic equation. The method employs an orthogonal polynomial collocation technique on multiple subdomains. The subdomain interfaces are approximated with the aid of a penalty method. The time discretization is implemented in an explicit/implicit finite difference method. The subdomain interface is approximated using an explicit Dufort-Frankel method, while the interior of each subdomain is approximated using an implicit backwards Euler's method. The principal advantage to the method is the direct implementation on a distributed computing system with a minimum of interprocessor communication. Theoretical results are given for Legendre polynomials, while computational results are given for Chebyshev polynomials. Results are given for both a single processor computer and a distributed computing system.