ESDG.jl (energy/entropy stable discontinuous Galerkin)
A Julia codebase for solving 1D/2D/3D time-dependent hyperbolic PDEs using energy or entropy stable high order DG methods on conforming unstructured meshes consisting of triangular, quadrilateral, or hexahedral elements.
These codes are inspired by the Matlab codes for the book Nodal Discontinuous Galerkin methods by Hesthaven and Warburton (2007). While unstructured meshes are supported, all demos use uniform meshes, and the codebase is intended mainly for experimentation and method development.
Demos for linear problems
- The simplest demo is dg1D_advec.jl, which solves the 1D advection equation on a uniform 1D mesh. The demo dg2D_advec_tri.jl simulates the advection equation on a triangular mesh. The setup of data structures on reference and physical elements is left exposed.
- The demos dg2D_wave_tri.jl, dg2D_wave_quad.jl compute solutions to the acoustic wave equation on triangular and quadrilateral meshes. The setup of data structures on reference and physical elements is hidden away in setup code.
- The demo dg3D_advec_hex.jl solves the advection equation on a hexahedral mesh.
Demos for nonlinear problems
- The files dg2D_euler_quad.jl and dg3D_euler_hex.jl provide entropy stable DG methods on quadrilateral and hexahedral meshes for the compressible Euler equations, with a sparsity-optimized implementation of the Hadamard sum step in flux differencing.
References
The discretizations used are based on the following references:
- Nodal discontinuous Galerkin methods
- Weight-adjusted discontinuous Galerkin methods: wave propagation in heterogeneous media
- On discretely entropy conservative and entropy stable DG methods
- Skew-Symmetric Entropy Stable Modal DG Formulations
- Efficient Entropy Stable Gauss Collocation Methods
Special thanks to Yimin Lin for providing the initial routines which started this codebase.