A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations

Abstract

In this contribution we derive and analyze a new numerical method for kinetic equations based on a coordinate transformation of the moment approximation. Classical minimum-entropy moment (M N ) closures are a class of reduced models for kinetic equations that conserve many of the fundamental physical properties of the solutions. However, their practical use is limited by their high-computational cost, as an optimization problem has to be solved for every cell in the space-time grid. In addition, implementation of numerical solvers for these models is hampered by the fact that the optimization problems are only well-defined if the moment vectors stay within the realizable set. For the same reason, further reducing these models by, e.g., reduced-basis methods is not a simple task. Our new method overcomes these disadvantages of classical approaches. The coordinate transformation is performed on the semi-discretized level which makes them applicable to a wide range of kinetic schemes and replaces the nonlinear optimization problems by inversion of the positive-definite Hessian matrix. As a result, the new scheme gets rid of the realizability-related problems. Moreover, our numerical experiments demonstrate that our new method is often several times faster than the standard optimization-based scheme.