Design of DIRK Schemes with High Weak Stage Order

TL;DR

This paper develops a general theory for high weak stage order in DIRK schemes, enabling the construction of high-order, stable, and efficient implicit methods that mitigate order reduction in stiff problems.

Contribution

It introduces a comprehensive framework for designing high weak stage order DIRK schemes beyond previous limits, improving stability and accuracy in stiff ODE and PDE problems.

Findings

Constructed DIRK schemes with WSO 4 and above.

Schemes are stiffly accurate and L-stable.

Demonstrated superior performance on test problems.

Abstract

Runge-Kutta (RK) methods may exhibit order reduction when applied to certain stiff problems. While fully implicit RK schemes exist that avoid order reduction via high-stage order, DIRK (diagonally implicit Runge-Kutta) schemes are practically important due to their structural simplicity; however, these cannot possess high stage order. The concept of weak stage order (WSO) can also overcome order reduction, and it is compatible with the DIRK structure. DIRK schemes of WSO up to 3 have been proposed in the past, however, based on a simplified framework that cannot be extended beyond WSO 3. In this work a general theory of WSO is employed to overcome the prior WSO barrier and to construct practically useful high-order DIRK schemes with WSO 4 and above. The resulting DIRK schemes are stiffly accurate, L-stable, have optimized error coefficients, and are demonstrated to perform well on a portfolio of relevant ODE and PDE test problems.