# Numerical Methods

Projects with a strong component in the development of numerical methods and numerical analysis.

### Krylov-Riemann Solver for Large Hyperbolic Conservation Laws

In this project a Riemann solver is derived for nonlinear hyperbolic systems of conservation
laws based on a Krylov subspace approximation of the upwinding dissipation vector. In the
general case, the solver does not require any detailed information of the eigensystem, except an estimate
of the global maximal propagation speed. It uses successive flux function evaluations to obtain
a numerical flux which is almost equivalent to that of a Godunov scheme with complete upwinding.
The new Krylov–Riemann solver is particularly efficient when used for large systems with many nonlinear
equations such that typically no explicit expression for the eigensystem is available. Also, no
numerical procedures are necessary to compute the eigensystem. Numerical examples demonstrate
the excellent performance of the solver with respect to other solvers.

**Contact:** Manuel Torrilhon.

### Fully Implicit Runge-Kutta Methods with Sparse Linear Solvers for PDEs

### Finite-Element Methods for Gas Micro-Flows

Since the regularized 13-Moment equations (R13) have become a promising contender in tackling the challenges of a rarefied gas-flow, a working numerical framework is desirable for computing solutions on complex domains. The R13 equation set shows a diffusive behavior for some flow situations which in turn makes finite elements the method of choice. But deriving a stable formulation in this case is not a straight forward task. In fact, challenges already arise when looking at a simplified linearized subsystem. The implementation of non-standard boundary conditions or the proper handling of a two-fold saddle point problem are only two examples for the variety of obstacles. Within this project, tasks ranging from numerical analysis to physical interpretation of simulation results are covered. To be more precise, a stable formulation for the non-linear equations shall be derived, efficiently implemented and tested against physical intuition (based on analytic solutions) and experimental data.

**Contact:** Armin Westerkamp and Manuel Torrilhon.