Communications in Mathematical Sciences

Volume 22 (2024)

Number 3

Spatial manifestations of order reduction in Runge–Kutta methods for initial boundary value problems

Pages: 613 – 653

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n3.a2

Authors

Rodolfo Ruben Rosales (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Benjamin Seibold (Department of Mathematics, Temple University, Philadelphia, Pennsylvania, U.S.A.)

David Shirokoff (Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, N.J., U.S.A.)

Dong Zhou (Department of Mathematics, California State University Los Angeles, Calif., U.S.A.)

Abstract

This paper studies the spatial manifestations of order reduction that occur when timestepping initial-boundary-value problems (IBVPs) with high-order Runge–Kutta methods. For such IBVPs, geometric structures arise that do not have an analog in ODE IVPs: boundary layers appear, induced by a mismatch between the approximation error in the interior and at the boundaries. To understand those boundary layers, an analysis of the modes of the numerical scheme is conducted, which explains under which circumstances boundary layers persist over many time steps. Based on this, two remedies to order reduction are studied: first, a new condition on the Butcher tableau, called weak stage order, that is compatible with diagonally implicit Runge–Kutta schemes; and second, the impact of modified boundary conditions on the boundary layer theory is analyzed.

Keywords

initial-boundary-value problem, time-stepping, Runge–Kutta, order reduction, boundary layer, stage order, weak stage order, modified boundary conditions

2010 Mathematics Subject Classification

34E05, 65L20, 65M15

Received 26 October 2019

Received revised 7 March 2022

Accepted 11 August 2023

Published 4 March 2024