AuthorsT. K. Nilssen, G. A. Staff and K. Mardal
TitleOrder Optimal Preconditioners for Fully Implicit Runge-Kutta Schemes Applied to the Bidomain Equations
AfilliationScientific Computing
Project(s)Center for Biomedical Computing (SFF)
StatusPublished
Publication TypeJournal Article
Year of Publication2011
JournalNumerical Methods for Partial Differential Equations
Volume27
Issue5
Pagination1290-1312
PublisherWiley Subscription Services, Inc., A Wiley Company
Abstract

The partial differential equation part of the bidomain equations is discretized in time with fully implicit Runge-Kutta methods, and the resulting block systems are preconditioned with a block diagonal preconditioner. By studying the time-stepping operator in the proper Sobolev spaces, we show that the preconditioned systems have bounded condition numbers given that the Runge-Kutta scheme is A-stable and irreducible with an invertible coefficient matrix. A new proof of order optimality of the preconditioners for the one-leg discretization in time of the bidomain equations is also presented. The theoretical results are verified by numerical experiments. Additionally, the concept of weakly positive-definite matrices is introduced and analyzed.

Notes

Listed in annual report 2010. Published online june 2010.

DOI10.1002/num.20582
Citation KeySimula.SC.35