A note on efficient preconditioner of implicit Runge–Kutta methods with application to fractional diffusion equations

摘要

This paper is concerned with the construction of efficient preconditioning method for systems arising from implicit Runge–Kutta discretizations of time-dependent PDEs. A polynomial preconditioner is proposed based on the Kronecker product splitting iteration method introduced by Chen (BIT 54:607-721, 2014). Some useful properties on the spectrum of the preconditioned matrix are established. Numerical experiments concerning discrete solution of fractional diffusion equations are presented to show the effectiveness of the proposed polynomial preconditioner.