Fourth-order paired-explicit Runge-Kutta methods

Computational Science and Engineering, vol. 2

Abstract

Abstract In this paper, we extend the Paired-Explicit Runge-Kutta (P-ERK) schemes by Vermeire et al. (J Comput Phys 393:465–483, 2019) and Nasab and Vermeire (J Comput Phys 468:111470, 2022) to fourth-order of consistency. Based on the order conditions for partitioned Runge-Kutta methods we motivate a specific form of the Butcher arrays which leads to a family of fourth-order accurate methods. The employed form of the Butcher arrays results in a special structure of the stability polynomials, which needs to be adhered to for an efficient optimization of the domain of absolute stability. We demonstrate that the constructed fourth-order P-ERK methods satisfy linear stability, internal consistency, designed order of convergence, and conservation of linear invariants. At the same time, these schemes are seamlessly coupled for codes employing a method-of-lines approach, in particular without any modifications of the spatial discretization. We demonstrate speedup for single-threaded program executions, shared-memory parallelism, i.e., multi-threaded executions and distributed-memory parallelism with MPI. We apply the multirate P-ERK schemes to inviscid and viscous problems with locally varying wave speeds, which may be induced by non-uniform grids or multiscale properties of the governing partial differential equation. Compared to state-of-the-art optimized standalone methods, the multirate P-ERK schemes allow significant reductions in right-hand-side evaluations and wall-clock time, ranging from $$66\%$$ up to factors greater than four. A reproducibility repository is provided which enables the reader to examine all results presented in this work.

Authors 5

1. Daniel Doehring corresponding Aachen Applied and Computational Mathematics

   RWTH Aachen University

   Affiliation as printed

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

2. Lars Christmann Aachen Applied and Computational Mathematics

   University of Cologne · RWTH Aachen University

   Affiliation as printed

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

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

3. 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, Aachen, Germany

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

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

4. Gregor J. Gassner

   University of Cologne

   Affiliation as printed

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

5. Manuel Torrilhon Aachen Applied and Computational Mathematics

   RWTH Aachen University

   Affiliation as printed

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