Chapter 2
Discontinuous Galerkin discretization

With the continuity equation given in its continuous form (1.5), solving this system of equations analytically has been accomplished only for special cases so far. Those special cases are e.g. one-dimensional simplifications and particular initial as well as boundary conditions [SES06,Syn91]. Therefore, we have to solve these equations numerically by discretization of the continuous form of the continuity equation.

Several approaches exist to solve such a system of equations numerically. Here, we give a short overview of the most important ones with an Eulerian [Lam32] approach in spatial domain:

• Finite differences belong to one of the most traditional methods. They approximate the spatial derivatives by computing derivatives based on particular points in the simulation domain. Following this approach, conserved quantities are given per point.
• Classical finite elements methods (FEM) discretize the equations based on an overlapping support of basis functions of disjunct cells. This leads to continuous approximated solutions at the cell boundaries. However, continuous finite element methods suffer from complex stencils with access patterns over several cells.
• With the Discontinuous Galerkin finite elements method (DG-FEM), a similar approach as for the classical FEM is taken. However, the basis functions are chosen in a way that their support is cell-local. On the one hand, this avoids complex access patterns, whereas on the other hand, this generates discontinuities at cell borders requiring solvers for the so called Riemann problem. Finite volume simulations can be interpreted as a special case of the DG-FEM formulation with a single basis function of 0-th order.

With our main focus on wave-propagation dominated problems and the DG-FEM being well-suited to solve such problems (see e.g.  [LeV02,HW08,Coc98]), we continue studying the discretization of the continuity equation for the remainder of this thesis with the DG-FEM. To determine the framework requirements to run such DG-FEM simulations on dynamic adaptive grids, a basic introduction to the discretization and approximations of the solution in each grid cell is required and given in the next sections.

 2.1 Grid generation
 2.2 Triangle reference and world space
 2.3 Basis functions
  2.3.1 Nodal basis functions
  2.3.2 Modal basis functions
 2.4 Weak formulation
 2.5 Mass matrix
 2.6 Stiffness matrices
 2.7 Flux matrices
 2.8 Source term
 2.9 Rotational invariancy and edge space
 2.10 Numerical flux
  2.10.1 Rusanov flux
  2.10.2 Limiter
 2.11 Boundary conditions
  2.11.1 Dirichlet & Inflow
  2.11.2 Outflow
  2.11.3 Bounce back
 2.12 Adaptive refining and coarsening matrices and
  2.12.1 Coefficient matrix
  2.12.2 Affine transformations to and from the child space
  2.12.3 Prolongation to child space
  2.12.4 Restriction to the parent space
  2.12.5 Adaptivity based on error indicators
 2.13 CFL stability condition
 2.14 Time stepping schemes