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<p>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.</p><p>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.</p><p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44207_2025_5_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(66\%\)</EquationSource> </InlineEquation> up to factors greater than four.</p><p>A reproducibility repository is provided which enables the reader to examine all results presented in this work.</p>

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Fourth-order paired-explicit Runge-Kutta methods

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年10月13日

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  • Daniel Doehring,
  • Lars Christmann,
  • Michael Schlottke-Lakemper,
  • Gregor J. Gassner,
  • Manuel Torrilhon

摘要

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.