An application of Discontinuous Galerkin space and velocity discretisations to the solution of a model kinetic equation

Abstract

An approach based on a Discontinuous Galerkin discretisation is proposed for the Bhatnagar–Gross–Krook model kinetic equation. This approach allows for a high-order polynomial approximation of molecular velocity distribution function both in spatial and velocity variables. It is applied to model one-dimensional normal shock wave and heat transfer problems. Convergence of solutions with respect to the number of spatial cells and velocity bins is studied, with the degree of polynomial approximation ranging from zero to four in the physical space variable and from zero to eight in the velocity variable. This approach is found to conserve mass, momentum and energy when high-degree polynomial approximations are used in the velocity space. For the shock wave problem, the solution is shown to exhibit accelerated convergence with respect to the velocity variable. Convergence with respect to the spatial variable is in agreement with the order of the polynomial approximation used. For the heat transfer problem, it was observed that convergence of solutions obtained by high-degree polynomial approximations is only second order with respect to the resolution in the spatial variable. This is attributed to the temperature jump at the wall in the solutions. The shock wave and heat transfer solutions are in excellent agreement with the solutions obtained by a conservative finite volume scheme.

Keywords:

• high-order numerical methods
• Discontinuous Galerkin methods
• model kinetic equations
• shock wave
• heat transfer

Acknowledgements

The authors thank Professor D. Arnold for insightful discussions of fast convergence of quadrature formulas and for his interest in this work. The authors are thankful to Professor L.L. Foster for her help in the preparation of the final version of the article. The first author was supported bythe 2009 ASEE Air Force Summer Faculty Fellowship Program.