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Multiderivative time integration methods preserving nonlinear functionals via relaxation

Hendrik Ranocha and Jochen Schütz

Vol. 19 (2024), No. 1, 27–56
Abstract

We combine the recent relaxation approach with multiderivative Runge–Kutta methods to preserve conservation or dissipation of entropy functionals for ordinary and partial differential equations. Relaxation methods are minor modifications of explicit and implicit schemes, requiring only the solution of a single scalar equation per time step in addition to the baseline scheme. We demonstrate the robustness of the resulting methods for a range of test problems including the 3D compressible Euler equations. In particular, we point out improved error growth rates for certain entropy-conservative problems including nonlinear dispersive wave equations.

Keywords
two-derivative methods, multiderivative methods, invariants, conservative systems, dissipative systems, structure-preserving methods
Mathematical Subject Classification
Primary: 65L06, 65M20, 65M70
Milestones
Received: 7 November 2023
Revised: 7 February 2024
Accepted: 17 March 2024
Published: 17 June 2024
Authors
Hendrik Ranocha
Institute of Mathematics
Johannes Gutenberg Universität Mainz
Mainz
Germany
Jochen Schütz
Faculty of Sciences & Data Science Institute
Hasselt University
Diepenbeek
Belgium