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

Communications on Applied Mathematics and Computation, vol. 7, pp. 3–39

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.

Authors 7

1. Hendrik Ranocha corresponding

   Universität Hamburg

   Affiliation as printed

   Applied Mathematics, University of Hamburg, Bundesstr. 55, 20146, Hamburg, Germany

2. Andrew Ross Winters

   Linköping University

   Affiliation as printed

   Department of Mathematics; Applied Mathematics, Linköping University, Linköping, Sweden

   Applied Mathematics, Linköping University, Linköping, Sweden

   Department of Mathematics

3. Hugo G. Castro

   King Abdullah University of Science and Technology

   Affiliation as printed

   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. Lisandro Dalcín

   King Abdullah University of Science and Technology

   Affiliation as printed

   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

5. Michael Schlottke‐Lakemper Aachen Applied and Computational Mathematics

   University of Stuttgart · RWTH Aachen University

   Affiliation as printed

   Applied and Computational Mathematics, RWTH Aachen University, Aachen, Germany

   High-Performance Computing Center Stuttgart, University of Stuttgart, Stuttgart, Germany

6. Gregor J. Gassner

   University of Cologne

   Affiliation as printed

   Center for Data and Simulation Science, University of Cologne, Cologne, Germany

   Department of Mathematics and Computer Science, University of Cologne, Cologne, Germany

7. Matteo Parsani

   King Abdullah University of Science and Technology

   Affiliation as printed

   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

   Physical Science and Engineering Division (PSE), King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia