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

TL;DR

This paper introduces a novel discretization method for kinetic equations that transforms the moment approximation to avoid complex optimization, leading to faster computations and better numerical stability.

Contribution

The authors develop a variable transformation-based discretization that replaces nonlinear optimization with matrix inversion, improving efficiency and realizability in minimum-entropy moment closures.

Findings

Method is several times faster than traditional approaches.

Eliminates realizability constraints in moment approximations.

Enforces discrete entropy law through modified time stepping.

Abstract

In this contribution we derive and analyze a new numerical method for kinetic equations based on a variable transformation of the moment approximation. Classical minimum-entropy moment closures are a class of reduced models for kinetic equations that conserve many of the fundamental physical properties of 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 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, a discrete entropy law can be enforced by modifying the time stepping scheme. Our numerical experiments demonstrate that our new method is often several times faster than the standard optimization-based scheme.