On Error-Based Step Size Control for Discontinuous Galerkin Methods for Compressible Fluid Dynamics

doi:10.1007/s42967-023-00264-y

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (1): 3-39.doi: 10.1007/s42967-023-00264-y

• ORIGINAL PAPERS • Previous Articles

On Error-Based Step Size Control for Discontinuous Galerkin Methods for Compressible Fluid Dynamics

Hendrik Ranocha¹, Andrew R. Winters², Hugo Guillermo Castro³, Lisandro Dalcin³, Michael Schlottke-Lakemper^4,5, Gregor J. Gassner^6,7, Matteo Parsani^3,8

1 Applied Mathematics, University of Hamburg, Bundesstr. 55, 20146 Hamburg, Germany;
2 Department of Mathematics;Applied Mathematics, Linköping University, Linköping, Sweden;
3 Extreme Computing Research Center (ECRC), Computer Electrical and Mathematical Science and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia;
4 Applied and Computational Mathematics, RWTH Aachen University, Aachen, Germany;
5 High-Performance Computing Center Stuttgart, University of Stuttgart, Stuttgart, Germany;
6 Department of Mathematics and Computer Science, University of Cologne, Cologne, Germany;
7 Center for Data and Simulation Science, University of Cologne, Cologne, Germany;
8 Physical Science and Engineering Division (PSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia

Received:2022-09-15 Revised:2023-02-02 Accepted:2023-02-15 Published:2025-04-21
Supported by:
Open Access funding enabled and organized by Projekt DEAL. Andrew Winters was funded through Vetenskapsrådet, Sweden Grant Agreement 2020-03642 VR. Some computations were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at Tetralith, partially funded by the Swedish Research Council under Grant Agreement No. 2018-05973. Hugo Guillermo Castro was funded through the award P2021-0004 of King Abdullah University of Science and Technology. Some of the simulations were enabled by the Supercomputing Laboratory and the Extreme Computing Research Center at King Abdullah University of Science and Technology. Gregor Gassner acknowledges funding through the Klaus-Tschira Stiftung via the project “HiFiLab”. Gregor Gassner and Michael Schlottke-Lakemper acknowledge funding from the Deutsche Forschungsgemeinschaft through the research unit “SNuBIC” (DFG-FOR5409).

Abstract

Abstract: We study a temporal step size control of explicit Runge-Kutta (RK) methods for compressible computational fluid dynamics (CFD), including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler equations. We demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy (CFL) number. Our numerical examples show that the error-based step size control is easy to use, robust, and efficient, e.g., for (initial) transient periods, complex geometries, nonlinear shock capturing approaches, and schemes that use nonlinear entropy projections. We demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases, the open source Julia packages Trixi.jl with OrdinaryDiffEq.jl and the C/Fortran code SSDC based on PETSc.

Key words: Explicit Runge-Kutta (RK) methods, Step size control, Compressible fluid dynamics, Adaptivity in space and time, Shock capturing

CLC Number:

Hendrik Ranocha, Andrew R. Winters, Hugo Guillermo Castro, Lisandro Dalcin, Michael Schlottke-Lakemper, Gregor J. Gassner, Matteo Parsani. On Error-Based Step Size Control for Discontinuous Galerkin Methods for Compressible Fluid Dynamics[J]. Communications on Applied Mathematics and Computation, 2025, 7(1): 3-39.

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On Error-Based Step Size Control for Discontinuous Galerkin Methods for Compressible Fluid Dynamics

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