Numerical methods for kinetic equations (Master's thesis)

Kinetic equations play an important role in many physical applications. Prominent examples are the Boltzmann equation of gas dynamics and the radiative transfer equation. In general, analytic solutions are not available and thus numerical solutions have to be found. However, due to their high dimensionality, kinetic equations cause a great amount of computational cost which may effectively make it impossible to get a sufficiently accurate solution using standard numerical solvers. Thus, methods that can find approximate solutions with less effort have to be used. A popular approach is to express the solution in terms of the first moments of the kinetic equation. This eliminates the dependency on the velocity variable and reduces the computational cost significantly. However, the resulting hyperbolic system of equations still has to be solved in several dimensions. Furthermore, many moments and thus a large system of equations may be necessary to get a reasonable approximation to the true solution. Hence, efficient solvers are still required to solve the problem in reasonable time. The goal of this thesis was the implementation of an efficient and generic solver for hyperbolic systems of equations in the C++ software framework DUNE. The implementation was tested against problems with known solution and existing solvers for the moment models