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References

This page lists background references for the numerical methods used by Exasim. It is not a complete bibliography, but it provides starting points for DG, LDG, HDG, implicit time integration, GMRES, preconditioning, and performance-portable GPU implementation.

DG And LDG

  • B. Cockburn and C.-W. Shu, "The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems", Journal of Computational Physics, 1998.
  • B. Cockburn and C.-W. Shu, "The local discontinuous Galerkin method for time-dependent convection-diffusion systems", SIAM Journal on Numerical Analysis, 1998.
  • B. Cockburn, G. E. Karniadakis, and C.-W. Shu, editors, Discontinuous Galerkin Methods: Theory, Computation and Applications, Springer, 2000.
  • N. C. Nguyen, S. Terrana, and J. Peraire, "Large-Eddy Simulation of Transonic Buffet Using Matrix-Free Discontinuous Galerkin Method", AIAA Journal, 60(5), 3060-3077, 2022. https://doi.org/10.2514/1.j060459.

HDG

  • B. Cockburn, J. Gopalakrishnan, and R. Lazarov, "Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second-order elliptic problems", SIAM Journal on Numerical Analysis, 2009.
  • B. Cockburn, J. Gopalakrishnan, and F.-J. Sayas, "A projection-based error analysis of HDG methods", Mathematics of Computation, 2010.
  • N. C. Nguyen, J. Peraire, and B. Cockburn, "An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations", Journal of Computational Physics, 2009.
  • N. C. Nguyen and J. Peraire, "Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics", Journal of Computational Physics, 231(18), 5955-5988, 2012. https://doi.org/10.1016/j.jcp.2012.02.033.

Time Integration

  • R. Alexander, "Diagonally implicit Runge-Kutta methods for stiff ODEs", SIAM Journal on Numerical Analysis, 1977.
  • E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer.
  • C. A. Kennedy and M. H. Carpenter, "Additive Runge-Kutta schemes for convection-diffusion-reaction equations", Applied Numerical Mathematics, 2003.

Krylov Solvers And Preconditioning

  • Y. Saad and M. H. Schultz, "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems", SIAM Journal on Scientific and Statistical Computing, 1986.
  • Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM.
  • A. Toselli and O. Widlund, Domain Decomposition Methods: Algorithms and Theory, Springer.
  • A. Welter and N. Cuong Nguyen, "Preconditioning techniques for Hybridizable discontinuous Galerkin discretizations on GPU architectures", Computer Methods in Applied Mechanics and Engineering, 456, 118951, 2026. https://doi.org/10.1016/j.cma.2026.118951.
  • P. Fernandez, N. C. Nguyen, and J. Peraire, "The hybridized Discontinuous Galerkin method for Implicit Large-Eddy Simulation of transitional turbulent flows", Journal of Computational Physics, 336, 308-329, 2017. https://doi.org/10.1016/j.jcp.2017.02.015.

GPU And Performance Portability

  • H. C. Edwards, C. R. Trott, and D. Sunderland, "Kokkos: Enabling manycore performance portability through polymorphic memory access patterns", Journal of Parallel and Distributed Computing, 2014.
  • C. R. Trott, D. Lebrun-Grandie, D. Arndt, and contributors, Kokkos documentation and publications on performance-portable C++ programming.
  • A. Welter and N. Cuong Nguyen, "Preconditioning techniques for Hybridizable discontinuous Galerkin discretizations on GPU architectures", Computer Methods in Applied Mechanics and Engineering, 456, 118951, 2026. https://doi.org/10.1016/j.cma.2026.118951.
  • J. Vila-Pérez, R. L. van Heyningen, N.-C. Nguyen, and J. Peraire, "Exasim: Generating discontinuous Galerkin codes for numerical solutions of partial differential equations on graphics processors", SoftwareX, 20, 101212, 2022. https://doi.org/10.1016/j.softx.2022.101212.
  • D. S. Hoskin, R. L. van Heyningen, N. C. Nguyen, J. Vila-Pérez, W. L. Harris, and J. Peraire, "Discontinuous Galerkin methods for hypersonic flows", Progress in Aerospace Sciences, 146, 100999, 2024. https://doi.org/10.1016/j.paerosci.2024.100999.

Exasim Internal Notes

The repository also contains implementation-focused internal notes under old/docs/03-internals/, including LDG implementation and block-diagonal Jacobian derivations. The web-facing versions are: