Overview

This package provides a simple implementation of Discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws. It is designed to be easy to understand and extend, making it suitable for educational purposes and quick prototyping of new ideas. It is not intended for high-performance computing or production use. Many ideas and concepts are inspired by the package Trixi.jl and most users will likely want to use that package instead. For the basic concepts of Discontinuous Galerkin methods, which are mostly the same as in Trixi.jl, please refer to the Trixi.jl documentation. The syntax of the API is very similar to Trixi.jl. Please check out the examples/ folder for some usage examples.

Differences to Trixi.jl

• The focus of SimpleDiscontinuousGalerkin.jl is on understandability and extendability of the code and less on performance and a wide variety of features. It is designed to closely follow the mathematical description of the DG method, which can make it suitable for teaching and learning purposes. For example, SimpleDiscontinuousGalerkin.jl implements more equivalent formulations like weak and strong formulation of the flux-differencing method to demonstrate the equivalence of these formulations.
• SimpleDiscontinuousGalerkin.jl has no support for many features Trixi.jl has (many more equations, multiple space dimensions, more sophisticated numerical fluxes, adaptive mesh refinement, shock capturing, positivity-preserving methods, limiting strategies, callbacks, parallel computing, etc.).
• SimpleDiscontinuousGalerkin.jl has some (rather niche) features Trixi.jl does not support (yet), which stem from quick tests and prototyping to test some new ideas. This includes different numerical fluxes at the boundary than in the interior and different bases for each element via the PerElementFDSBP solver.
• SimpleDiscontinuousGalerkin.jl has support for one-dimensional overset grid (Chimera) methods.
• In contrast to Trixi.jl, which implements an own Gauss-Lobatto-Legendre basis for the Discontinuous Galerkin spectral element method (DGSEM), this package highlights more the underlying summation-by-parts (SBP) structure of the DG method. It reuses the legendre_derivative_operator from SummationByPartsOperators.jl for the Discontinuous Galerkin spectral element method (DGSEM). This can be seen as a special case of FDSBP with the underlying SBP operator being the (Gauss-Lobatto-)Legendre derivative operator.