Order optimal preconditioners for fully implicit Runge-Kutta schemes applied to the bidomain equations
✍ Scribed by Trygve K. Nilssen; Gunnar A. Staff; Kent-Andre Mardal
- Publisher
- John Wiley and Sons
- Year
- 2010
- Tongue
- English
- Weight
- 869 KB
- Volume
- 27
- Category
- Article
- ISSN
- 0749-159X
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✦ Synopsis
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.