Discontinuous Galerkin for solving conservation law.

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

Four vortex sheets Density and Mesh | Muzzle break Density and Mesh |