Structured derivation of moment equations and stable boundary conditions with an introduction to symmetric, trace-free tensors

Abstract

Moment equations are used to approximate kinetic equations in a physically meaningful and numerically efficient way. This paper discusses the derivation of general moment equations including various caveats that must be taken care of when moment systems get implemented in simulation tools. We suggest to use a constant Maxwellian with arbitrary constant reference values for density, velocity and temperature for the underlying expansion of the distribution. This seems restrictive in comparison to using a local Maxwellian, but full nonlinear fluid dynamics remains possible as long as the reference values do not differ too much from the local values and a sufficiently large moment vector is considered. Additionally, this approach allows to derive globally hyperbolic equations in a simple way and the transport operator becomes linear. The resulting equations are discussed in detail and compared to various existing approaches.The paper also presents boundary conditions that render the moment systems stable in an $ L^{2} $-sense. This is done both by general considerations and for the concrete case of the Maxwell accommodation model for kinetic equations, generalizing earlier stable boundary conditions for special cases of moment equations. The extensive appendix of the paper will help readers unfamiliar with tensor variables and symmetric, trace-free decompositions to understand the general concept needed for moment approximations without the need of group-theoretical expertise.