|Authors||T. K. Nilssen, G. A. Staff and K. Mardal|
|Title||Order Optimal Preconditioners for Fully Implicit Runge-Kutta Schemes Applied to the Bidomain Equations|
|Project(s)||Center for Biomedical Computing (SFF)|
|Publication Type||Journal Article|
|Year of Publication||2011|
|Journal||Numerical Methods for Partial Differential Equations|
|Publisher||Wiley Subscription Services, Inc., A Wiley Company|
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.
Listed in annual report 2010. Published online june 2010.