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RESEARCH PRODUCT

Order optimal preconditioners for fully implicit Runge-Kutta schemes applied to the bidomain equations

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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. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq Eq 27: 1290–1312, 2011