Inexact shift-invert Arnoldi method for evolution equations

Yuka Hashimoto, Takashi Nodera


Linear and nonlinear evolution equations with a first order time derivative, such as the heat equation, the Burgers equation, and the reaction diffusion equation have been used to solve problems in various fields of science. Differential algebraic equations of the first order are derived after space discretization. In the simplest case, the computation of one matrix exponential with a special form is required. In the most complex case, the computation of matrix functions related to its exponentials need to be implemented repeatedly. When computing large matrix functions, the Krylov subspace methods is a viable alternative. The most well-known method is the Arnoldi method, but it may require a number of iterations depending on the condition of the matrix. As a solution to this issue, we propose the Inexact Shift-invert Arnoldi method to do this more efficiently. As the result, the numerical solution of evolution equations can be computed efficiently with this method. Pertinent numerical experiments establish the effectiveness of this proposed algorithm.



Shift-invert Arnoldi; phi-function; exponential integrator

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ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.