DIRK Schemes with High Weak Stage Order

TL;DR

This paper introduces DIRK schemes with high weak stage order to prevent order reduction in stiff problems, expanding the applicability of DIRK methods with improved convergence properties.

Contribution

It proposes a weak stage order criterion compatible with DIRK schemes, and constructs specific DIRK methods with weak stage order up to 3.

Findings

DIRK schemes with weak stage order up to 3 are feasible.

The new schemes demonstrate improved performance on stiff problems.

Order reduction is mitigated using the proposed weak stage order approach.

Abstract

Runge-Kutta time-stepping methods in general suffer from order reduction: the observed order of convergence may be less than the formal order when applied to certain stiff problems. Order reduction can be avoided by using methods with high stage order. However, diagonally-implicit Runge-Kutta (DIRK) schemes are limited to low stage order. In this paper we explore a weak stage order criterion, which for initial boundary value problems also serves to avoid order reduction, and which is compatible with a DIRK structure. We provide specific DIRK schemes of weak stage order up to 3, and demonstrate their performance in various examples.