The Discontinuous Galerkin Method (DGM) was initially introduced by Reed and Hill in 1973 as a technique to solve neutron transport problems. Lesaint presented the first numerical analysis of the method for a linear advection equation. However, the technique lay dormant for several years and has only recently become popular as a method for solving fluid dynamics or electromagnetic problems. The DGM is somewhere between a finite element and a finite volume method and has many good features of both.
Our aim is to solve conservation laws i.e. problems that are written as:
where the unknown u has m components:
With the aim of constructing a Galerkin approximation of the conservation law, we integrate it on the domain, multiply by a family of test functions w, integrate by parts and equated to 0 for all w:
We introduce a discretization of the domain, a mesh
and a piecewise discontinuous approximation for u. We write then the conservation for each element e of the mesh:
Because the approxmation is discontinuous, the value of u on inter-element boundaries is not defined. We borrow here the solutions provided by high resolution finite difference and finite volume schemes by introducing a numerical flux at the interface that is calculated as the solution (or an approximate solution) of a Riemann problem:
Various choice of numerical fluxes are available: Roe linearization or Godounov exact 1D solution for strong schocks.
Some examples may be found at this link
More infos about DG can be found on this paper or this other paper.