# SC²S Colloquium - August 6, 2014

### From Sccswiki

Date: |
August 6, 2014 |

Room: |
02.07.023 |

Time: |
3 pm, s.t. |

## Leonhard Rannabauer: Runge-Kutta and ADER discontinuous Galerkin schemes for hyperbolic partial differential equations

Additionally to the widely used Finite volume and continuous Galerkin methods, discontinuous Galerkin methods provide a set of tools to numerically solve partial differential equations. In advantage to finite volume methods they offer higher-order accuracy. Compared to continuous Galerkin methods mass and stiffness matrices can be kept small, due to their local definition, where single elements only communicate with their direct neighbors. A common way to solve the time integrator of discontinuous Galerkin methods is to use Runge-Kutta solvers for ordinary differential equations. The order of convergence of these solvers is bound through the butcher barrier, making a higher order convergence hardly realizable.

To allow higher orders of convergence an alternative approach has been proposed, translating the principle of using the weak form of a partial differential equation, from the space dimension to the time dimension. The resulting discontinuous Galerkin time predictor scheme is similar to the semi discrete discontinuous Galerkin scheme and provides a new set of mass, stiffness and flux matrices.

In this presentation I will give an introduction to the ADER approach and compare it to the well known finite volume and the discontinuous Galerkin methods with Runge-Kutta time stepping and discuss the sparsity patterns of the matrices of these methods.