Stage-Parallel Preconditioners for Implicit Runge-Kutta Methods of Arbitrary High Order. Linear problems

Owe Axelsson, Ivo Dravins, and Maya Neytcheva

April 2022

Abstract:

Fully implicit Runge-Kutta methods offer the possibility to use high order accurate time discretization to match space discretization accuracy, an issue of significant importance for many large scale problems of current interest, where we may have fine space resolution with many millions of spatial degrees of freedom and long time intervals.

In this work we consider strongly A-stable implicit Runge-Kutta methods of arbitrary order of accuracy, based on Radau quadratures. For the arising large algebraic systems we introduce an efficient preconditioner, that allows for fully stage-parallel solution. We analyse the spectrum of the corresponding preconditioned system and illustrate the performance of the solution method with numerical experiments using MPI. In this work we consider only linear problems.

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