ALGEBRAIC STRUCTURE OF THE WEAK STAGE ORDER CONDITIONS FOR RUNGE-KUTTA METHODS

Abhijit Biswas, David Ketcheson, Benjamin Seibold, David Shirokoff

Research output: Contribution to journalArticlepeer-review

Abstract

Runge-Kutta (RK) methods may exhibit order reduction when applied to stiff problems. For linear problems with time-independent operators, order reduction can be avoided if the method satisfies certain weak stage order (WSO) conditions, which are less restrictive than traditional stage order conditions. This paper outlines the first algebraic theory of WSO, and establishes general order barriers that relate the WSO of a RK scheme to its order and number of stages for both fully-implicit and DIRK schemes. It is shown in several scenarios that the constructed bounds are sharp. The theory characterizes WSO in terms of orthogonal invariant subspaces and associated minimal polynomials. The resulting necessary conditions on the structure of RK methods with WSO are then shown to be of practical use for the construction of such schemes.

Original languageEnglish (US)
Pages (from-to)48-72
Number of pages25
JournalSIAM Journal on Numerical Analysis
Volume62
Issue number1
DOIs
StatePublished - 2024

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Keywords

  • B-convergence
  • DIRK methods
  • order-reduction
  • Runge-Kutta
  • weak stage order

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