Table of Contents Author Guidelines Submit a Manuscript
Advances in Mathematical Physics
Volume 2015, Article ID 787198, 13 pages
http://dx.doi.org/10.1155/2015/787198
Research Article

The Interaction of Iteration Error and Stability for Linear Partial Differential Equations Coupled through an Interface

1Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
2Department of Statistics, Colorado State University, Fort Collins, CO 80523, USA
3Tech X Corporation, Boulder, CO 80303, USA

Received 21 October 2014; Revised 16 December 2014; Accepted 18 December 2014

Academic Editor: Soheil Salahshour

Copyright © 2015 B. Sheehan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate properties of algorithms that are used to solve coupled evolutionary partial differential equations posed on neighboring, nonoverlapping domains, where the solutions are coupled by continuity of state and normal flux through a shared boundary. The algorithms considered are based on the widely used approach of iteratively exchanging boundary condition data on the shared boundary at each time step. There exists a significant and sophisticated numerical analysis of such methods. However, computations for practical applications are often carried out under conditions under which it is unclear if rigorous results apply while relatively few iterations are used per time step. To examine this situation, we derive exact matrix expressions for the propagation of the error due to incomplete iteration that can be readily evaluated for specific discretization parameters. Using the formulas, we show that the universal validity of several tenants of the practitioner’s conventional wisdom are not universally valid.