On Reusing the Stages of a Rejected Runge-Kutta Step

Vladislav N. Kovalnogov, Ruslan V. Fedorov, Tamara V. Karpukhina, Theodore E. Simos () and Charalampos Tsitouras
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Vladislav N. Kovalnogov: Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
Ruslan V. Fedorov: Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
Tamara V. Karpukhina: Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
Theodore E. Simos: Laboratory of Inter-Disciplinary Problems of Energy Production, Ulyanovsk State Technical University, 32 Severny Venetz Street, 432027 Ulyanovsk, Russia
Charalampos Tsitouras: General Department, National & Kapodistrian University of Athens, Euripus Campus, 34400 Evia, Greece

Mathematics, 2023, vol. 11, issue 11, 1-10

Abstract: Runge-Kutta (RK) pairs are amongst the most popular methods for numerically solving Initial Value Problems. While using an RK pair, we may experience rejection of some steps through the interval of integration. Traditionally, all of the evaluations are then dropped, and we proceed with a completely new round of computations. In this work, we propose avoiding this waste and continuing by reusing the rejected RK stages. We focus especially on an RK pair of orders six and five. After step rejection, we reuse all the previously evaluated stages and only add three new stages. We proceed by evaluating the output using a smaller step. By this technique, we manage to significantly reduce the cost in a set of problems that are known to pose difficulties in RK algorithms with changing step sizes.

Keywords: initial value problem; Runge-Kutta; step control (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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