Explicit Runge Kutta Methods that Alleviate Order Reduction

TL;DR

This paper introduces weak stage order conditions for explicit Runge-Kutta methods to prevent order reduction in linear and nonlinear initial-boundary value problems, enhancing convergence accuracy.

Contribution

It establishes a theoretical relationship between order, weak stage order, and stages, and develops explicit RK methods with high weak stage order to avoid order reduction.

Findings

High weak stage order methods prevent order reduction in linear problems.

Numerical tests confirm effectiveness up to any order for linear problems.

Methods also mitigate order reduction up to order three for nonlinear problems.

Abstract

Explicit Runge--Kutta (RK) methods are susceptible to a reduction in the observed order of convergence when applied to initial-boundary value problem with time-dependent boundary conditions. We study conditions on explicit RK methods that guarantee high-order convergence for linear problems; we refer to these conditions as weak stage order conditions. We prove a general relationship between the method's order, weak stage order, and number of stages. We derive explicit RK methods with high weak stage order and demonstrate, through numerical tests, that they avoid the order reduction phenomenon up to any order for linear problems and up to order three for nonlinear problems.