SPATIAL MANIFESTATIONS OF ORDER REDUCTION IN RUNGE-KUTTA METHODS FOR INITIAL BOUNDARY VALUE PROBLEMS*

Document Type

Article

Publication Date

1-1-2024

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.

Identifier

85189241805 (Scopus)

Publication Title

Communications in Mathematical Sciences

External Full Text Location

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

e-ISSN

19450796

ISSN

15396746

First Page

613

Last Page

653

Issue

3

Volume

22

Grant

DMS–2012268

Fund Ref

National Science Foundation

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