[Back]

G. Kitzler:
"A High Order Discontinuous Galerkin Method for the Boltzmann Equation";
Supervisor, Reviewer: J. Schöberl, Ch. Schmeiser; Institut für Analysis und Scientific Computing, 2016; oral examination: 2016-01-19.

In the underlying thesis, we present a numerical method to solve for the behaviour of a dilute gas. The mathematical model behind such a gas is the Boltzmann equation. It´s solution, usually denoted by f = f(t; x; v) depends on time, position and velocity and holds the average number of particles having position close to x and velocity close to v.
The discretization presented in the sequel is a Petrov-Galerkin projection. To numerically approximate the solution, which is defined in R7, we use a tensor product. The test functions are global polynomials in the velocity variable and local, discontinuous,
piecewise polynomials in the position variable. These test functions yield the conservation of physically conserved properties naturally. The trial functions are similar to the test functions chosen as discontinuous, piecewise polynomials in the spatial variable. In
the velocity variable, we take an approach of global polynomials multiplied with Gaussian peaks. This gives good approximation properties of solutions close to equilibrium and thus, close to the fluid regime.
The discontinuities of the trial and test functions yield skeleton integrals in the variational formulation. By choosing natural upwind fluxes in these skeleton integrals a stable discretization is achieved.
The Gaussian peak in the trial functions ensures additionally that all integrals over the unbounded momentum domain exist. For the evaluation we use the Gauss Hermite quadrature rules. In contrast to many other deterministic methods there is no additional modelling error due to domain truncation.
We extend the main idea in the following way: we shift and scale the Gaussian peaks element wise according to the gas´ local mean velocity and temperature calculated from the previous time step. The approximation properties of the trial space are greatly enhanced
by such a dependency on the solution. On the other hand, stability is decreased.
By smoothing the above mentioned parameters mean velocity and temperature slightly, the stability issue can be avoided for the most part.
The evaluation of the collision integrals in actual computations is a critical part since this involves a lot of numerical work. To reduce the complexity in the calculations we transform the solution from nodal to hierarchical polynomials to arrive at an inner integral
operator in diagonal form. We show how to use the properties of the trial spaces to execute this transformations efficiently.
Finally we show a lot of numerical examples as a validation for the method. This includes space homogeneous as well as space dependent problems. The results demonstrate the iii
excellent approximation properties of the shifted and scaled basis functions, especially close to the fluid regime. In addition, the computation times show the speed up achieved by the evaluation techniques for the collision integral.