Efficient Tensor-Product Spectral-Element Operators with the Summation-by-Parts Property on Curved Triangles and Tetrahedra
This repository contains the Julia code to reproduce the results in the following manuscript:
T. Montoya and D. W. Zingg, "Efficient Tensor-Product Spectral-Element Operators with the Summation-by-Parts Property on Curved Triangles and Tetrahedra," arXiv:2306.05975, 2024.
Please cite the above manuscript if you use this repository or the underlying spectral-element framework StableSpectralElements.jl in your research.
We present an extension of the summation-by-parts (SBP) framework to tensor-product spectral-element operators in collapsed coordinates. The proposed approach enables the construction of provably stable discretizations of arbitrary order which combine the geometric flexibility of unstructured triangular and tetrahedral meshes with the efficiency of sum-factorization algorithms. Specifically, a methodology is developed for constructing triangular and tetrahedral spectral-element operators of any order which possess the SBP property (i.e. satisfying a discrete analogue of integration by parts) as well as a tensor-product decomposition. Such operators are then employed within the context of discontinuous spectral-element methods based on nodal expansions collocated at the tensor-product quadrature nodes as well as modal expansions employing Proriol-Koornwinder-Dubiner polynomials, the latter approach resolving the time step limitation associated with the singularity of the collapsed coordinate transformation. Energy-stable formulations for curvilinear meshes are obtained using a skew-symmetric splitting of the metric terms, and a weight-adjusted approximation is used to efficiently invert the curvilinear modal mass matrix. The proposed schemes are compared to those using non-tensorial multidimensional SBP operators, and are found to offer comparable accuracy to such schemes in the context of smooth linear advection problems on curved meshes, but at a reduced computational cost for higher polynomial degrees.
First, make sure to install Julia if you haven't already done so. The tests in this paper were run on v1.8.5, and we recommend using that version or a later one. Earlier versions may result in incorrect operation counts due to this issue with diagonal matrix multiplication. Then, assuming that you are using Linux or macOS and have git installed, follow the steps below.
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Clone this repository by entering the command
git clone https://github.com/tristanmontoya/ReproduceSBPSimplex.gitin the terminal. -
Within the top-level
ReproduceSBPSimplexdirectory, use the commandjulia --project=.to open the Julia REPL and activate the project within the current directory. -
Install all dependencies by entering
using Pkg; Pkg.instantiate()in the REPL. This will automatically set up the latest version of StableSpectralElements.jl for you to use within this project.
Here, we describe how to generate the results using the provided scripts, and how to produce the results in the manuscript using the provided Jupyter notebooks. Note that some of the tests run a lot faster with multithreading enabled (for example, add --threads 8 to the julia command if you want to use eight threads). If using multiple Julia threads, it is usually best to set the number of BLAS threads to 1 (for example, using the OPENBLAS_NUM_THREADS environment variable). The table below lists the scripts containing the appropriate calls to the driver file for each numerical experiment, as well as the notebooks used to postprocess the simulation results in order to generate the figures. If a directory is listed in the scripts column, then all scripts in the directory should be run (in any order).
| Description | Figure | Script(s) | Postprocessing notebooks |
|---|---|---|---|
| Conservation and energy stability | 2 | N/A | notebooks/stability_conservation_tri.ipynb notebooks/stability_conservation_tet.ipynb |
| Spectral radius | 3 | scripts/sr_tri/ scripts/sr_tet/ |
notebooks/spectral_radius_tri.ipynb notebooks/spectral_radius_tet.ipynb |
| Accuracy | 4 | scripts/h_refine_tri/ scripts/p_refine_tri/ scripts/h_refine_tet/ scripts/p_refine_tet/ |
notebooks/convergence_plots_tri.ipynb notebooks/convergence_plots_tet.ipynb |
| Operation count | 5 | scripts/flops_tri.jl scripts/flops_tet.jl |
notebooks/flops_tri.ipynb notebooks/flops_tet.ipynb |
The data files directly used to generate the figures in the manuscript are provided in the results directory in HDF5-compatible JLD2 format, although the raw simulation datasets (which can be produced by running the above scripts) are not provided due to their size. Further inquiries regarding the code, results, and manuscript should be directed to tristan.montoya@mail.utoronto.ca.
This software is released under the GPLv3 license.