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Generating Discontinuous Galerkin Codes For Extreme Scalable Simulations

Exasim is an open-source software for generating high-order discontinuous Galerkin (DG) codes to numerically solve parametrized partial differential equations (PDEs) on different computing platforms with distributed memory. It combines high-level languages and low-level languages to easily construct parametrized PDE models and automatically produce high-performance C++ codes. The construction of parametrized PDE models and the generation of the stand-alone C++ production code are handled by high-level languages, while the production code itself can run on various machines, from laptops to the largest supercomputers, with both CPU and Nvidia GPU processors.

What make Exasim unique are the following distinctive features:

  • Solve a wide variety of PDEs in fluid and solid mechanics, and electromagnetism
  • Generate stand-alone C++ production code via the mathematical expressions of the PDEs
  • Implement local DG and hybridized DG methods for spatial discretization
  • Implement diagonally implicit Runge-Kutta methods for temporal discretization
  • Implement parallel Newton-GMRES solvers and scalable preconditioners using reduced basis method and polynomial preconditioners.
  • Employ Kokkos to provide full GPU functionality for all code components from discretization schemes to iterative solvers
  • Leverage Enzyme for automatic differentiation and Mutation++ for thermodynamic, transport, chemistry, and energy transfer properties.
  • Provide interfaces to Julia, Python, and Matlab.

After downloading the source code, please make sure that the name of the folder is Exasim. If it has a different name, please rename it to Exasim. Please make sure that the directory containing the folder Exasim does not have any white space, because Kokkos libraries can not be compiled properly in such case. See the documentation for more details.

Installation

Exasim needs Kokkos (required), Blas/Lapack libaries (required), MPI library (required), Gmesh for mesh generation (optional), METIS for mesh partitioning (optional), Paraview for visualization (optional), and CUDA Toolkit (optional) to run on Nvidia GPUs. These external packages can be installed by running install.jl in Julia, install.py in Python, or install.m in Matlab.

As Exasim generates and compiles stand-alone C++ code on the fly, Exasim does not require installation. However, since Exasim uses Kokkos to target various computing platforms, you must build Kokkos libraries before using Exasim. To build Kokkos serial library for CPU platform, please follow the below steps

  $ cd Exasim/kokkos   
  $ mkdir buildserial
  $ cd buildserial
  $ cmake .. -DCMAKE_INSTALL_PREFIX=../buildserial
  $ make install

To build Kokkos CUDA library for Nvidia GPU platform, please follow the below steps

  $ cd Exasim/kokkos
  $ mkdir buildcuda
  $ cd buildcuda
  $ cmake .. -DCMAKE_CXX_COMPILER=clang++ -DKokkos_ENABLE_CUDA=ON -DCMAKE_INSTALL_PREFIX=../buildcuda
  $ make install

Once Kokkos libraries are successfully built, you can start using Exasim. To try out any of the provided examples, please go to any folder in the directory Exasim/examples and run pdeapp.jl in Julia, pdeapp.py in Python, or pdeapp.m in Matlab.

Examples

Exasim produces C++ Code to solve a wide variety of parametrized partial differential equations from first-order, second-order elliptic, parabolic, hyperbolic PDEs, to higher-order PDEs. Many examples are provided in Exasim/examples to illustrate how to use Exasim for solving Poisson equation, wave equation, heat equation, advection, convection-diffusion, Euler equations, Navier-Stokes equations, and MHD equations. See the Bickley Jet example for simulation results.

To run any example with Julia, type the following line and hit return

   julia> include("pdeapp.jl")

To run any example with Python, type the following line and hit return

   > > > exec(open("pdeapp.py").read())

To run any example with Matlab, type the following line and hit return

   > >  pdeapp

If successful, Exasim produces an executable application and three new folders in the build folder. The build/model folder contains the Kokkos source code generated by Exasim, the build/datain folder contains input files for the executable application, and the build/dataout folder contains the output files produced by running the executable application, which stores the numerical solution of the PDE model defined in the pdeapp script. The name of the executable application is cpuEXASIM for CPU platform on one processor, cpumpiEXASIM for CPU platform on many processors, gpuEXASIM for CUDA platform on one GPU, and gpumpiEXASIM for CUDA platform on many GPUs.

Publications

[1] Vila-Pérez, J., Van Heyningen, R. L., Nguyen, N.-C., & Peraire, J. (2022). Exasim: Generating discontinuous Galerkin codes for numerical solutions of partial differential equations on graphics processors. SoftwareX, 20, 101212. https://doi.org/10.1016/j.softx.2022.101212

[2] Hoskin, D. S., Van Heyningen, R. L., Nguyen, N. C., Vila-Pérez, J., Harris, W. L., & Peraire, J. (2024). Discontinuous Galerkin methods for hypersonic flows. Progress in Aerospace Sciences, 146, 100999. https://doi.org/10.1016/j.paerosci.2024.100999

[3] Nguyen, N. C., Terrana, S., & Peraire, J. (2022). Large-Eddy Simulation of Transonic Buffet Using Matrix-Free Discontinuous Galerkin Method. AIAA Journal, 60(5), 3060–3077. https://doi.org/10.2514/1.j060459

[4] Nguyen, N. C., & Peraire, J. (2012). Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics. Journal of Computational Physics, 231(18), 5955–5988. https://doi.org/10.1016/j.jcp.2012.02.033