paper-2024-perk4

This repository contains information and code to reproduce the results presented in the article bibtex @online{doehring2024fourth, title={Fourth-Order Paired-Explicit Runge-Kutta Methods}, author={Doehring, Daniel and Christmann, Lars and Schlottke-Lakemper, Michael and Gassner, Gregor J. and Torrilhon, Manuel}, year={2024}, eprint={2408.05470}, eprinttype={arxiv}, eprintclass={math.NA}, url={https://arxiv.org/abs/2408.05470}, journal={arXiv preprint arXiv:2408.05470}, doi={10.48550/arXiv.2408.05470} }

If you find these results useful, please cite the article mentioned above. If you use the implementations provided here, please also cite this repository as bibtex @misc{doehring2024fourthRepro, title={Reproducibility repository for "Fourth-Order Paired-Explicit Runge-Kutta Methods"}, author={Doehring, Daniel and Christmann, Lars and Schlottke-Lakemper, Michael and Gassner, Gregor J. and Torrilhon, Manuel}, year={2024}, howpublished={\url{https://github.com/DanielDoehring/paper-2024-perk4}}, doi={https://doi.org/10.5281/zenodo.13899247} }

Abstract

In this paper, we extend the Paired-Explicit Runge-Kutta (P-ERK) schemes by Vermeire et. al. [1,2] 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. 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 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.

Reproducing the results

Installation

To download the code using git, use

bash git clone git@github.com:DanielDoehring/paper-2024-perk4.git

If you do not have git installed you can obtain a .zip and unpack it: bash wget https://github.com/DanielDoehring/paper-2024-perk4/archive/refs/heads/main.zip unzip main.zip mv paper-2024-perk4-main/ paper-2024-perk4

To instantiate the Julia environment execute the following two commands: bash cd paper-2024-perk4/paper-2024-perk4-main/ julia --project=. -e 'using Pkg; Pkg.instantiate()'

Note that the results are obtained using Julia 1.10.4, which is also set in the Manifest.toml. Thus, you might need to install the old Julia 1.10.4 release first and replace the julia calls from this README with /YOUR/PATH/TO/julia-1.10.4/bin/julia

Project initialization

If you installed Trixi.jl this way, you always have to start Julia with the --project flag set to your paper-2024-perk4 directory, e.g., bash julia --project=. if already inside the paper-2024-perk4 directory.

If you do not execute from the paper-2024-perk4 directory, you have to call julia with bash julia --project=/YOUR/PATH/TO/paper-2024-perk4

Running the code

The scripts for validations and applications are located in the 5_Validation and 6_Applications directory, respectively.

To execute them provide the respective path:

bash julia --project=. ./5_Validation/5_1_LinearStability/elixir_advection_linear_stability.jl

For all cases in the applications directory the solution has been computed using a specific number of threads. To specify the number of threads the --threads flag needs to be specified, i.e., bash julia --project=. --threads 24 ./6_Applications/6_4_SD7003Airfoil/Integrators_PERK4.jl The number of threads used for the examples are given in the README.md in 6_Applications.

Authors

• Daniel Doehring (Corresponding Author)
• Lars Christmann
• Michael Schlottke-Lakemper
• Gregor J. Gassner
• Manuel Torrilhon

Note that the Trixi authors are listed separately here.

Disclaimer

Everything is provided as is and without warranty. Use at your own risk!