These html pages are based on the PhD thesis "Cluster-Based Parallelization of Simulations on Dynamically Adaptive Grids and Dynamic Resource Management" by Martin Schreiber.
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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:

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 M
 2.6 Stiffness matrices S
 2.7 Flux matrices E
 2.8 Source term
 2.9 Rotational invariancy and edge space
 2.10 Numerical flux F
  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 R and C
  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