Order‐Optimal Preconditioners for Implicit Runge–Kutta Schemes Applied to Parabolic PDEs

Abstract

In this paper we show that standard preconditioners for parabolic PDEs discretized by implicit Euler or Crank–Nicolson schemes can be reused for higher‐order fully implicit Runge–Kutta time discretization schemes. We prove that the suggested block diagonal preconditioners are order‐optimal for A‐stable and irreducible Runge–Kutta schemes with invertible coefficient matrices. The theoretical investigations are confirmed by numerical experiments.