Multirate time-integration based on dynamic ODE partitioning through adaptively refined meshes for compressible fluid dynamics

Journal of Computational Physics, vol. 514, pp. 113223

Abstract

In this paper, we apply the Paired-Explicit Runge-Kutta (P-ERK) schemes by Vermeire et al. [1], [2] to dynamically partitioned systems arising from adaptive mesh refinement. The P-ERK schemes enable multirate time-integration with no changes in the spatial discretization methodology, making them readily implementable in existing codes that employ a method-of-lines approach. We show that speedup compared to a range of state of the art Runge-Kutta methods can be realized, despite additional overhead due to the dynamic re-assignment of flagging variables and restricting nonlinear stability properties. The effectiveness of the approach is demonstrated for a range of simulation setups for viscous and inviscid convection-dominated compressible flows for which we provide a reproducibility repository. In addition, we perform a thorough investigation of the nonlinear stability properties of the Paired-Explicit Runge-Kutta schemes regarding limitations due to the violation of monotonicity properties of the underlying spatial discretization. Furthermore, we present a novel approach for estimating the relevant eigenvalues of large Jacobians required for the optimization of stability polynomials.

Authors 4

1. Daniel Doehring corresponding Aachen Applied and Computational Mathematics

   RWTH Aachen University

   Affiliation as printed

   Applied and Computational Mathematics, RWTH Aachen University, Schinkelstrasse 2, Aachen, 52062, North Rhine-Westphalia, Germany

2. Michael Schlottke‐Lakemper Aachen Applied and Computational Mathematics

   University of Stuttgart · University of Augsburg · RWTH Aachen University

   Affiliation as printed

   Applied and Computational Mathematics, RWTH Aachen University, Schinkelstrasse 2, Aachen, 52062, North Rhine-Westphalia, Germany

   High-Performance Computing Center Stuttgart (HLRS), University of Stuttgart, Germany

   High-Performance Scientific Computing, Center for Advanced Analytics and Predictive Sciences, University of Augsburg, Germany

3. Gregor J. Gassner

   University of Cologne

   Affiliation as printed

   Department of Mathematics and Computer Science, Center for Data and Simulation Science, University of Cologne, Germany

4. Manuel Torrilhon Aachen Applied and Computational Mathematics

   RWTH Aachen University

   Affiliation as printed

   Applied and Computational Mathematics, RWTH Aachen University, Schinkelstrasse 2, Aachen, 52062, North Rhine-Westphalia, Germany