A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension

摘要

We implement a high-order numerical scheme for the entropy-based momentclosure, the so-called M$_N$ model, for linear kinetic equations in slabgeometry. A discontinuous Galerkin (DG) scheme in space along with astrong-stability preserving Runge-Kutta time integrator is a natural choice toachieve a third-order scheme, but so far, the challenge for such a scheme inthis context is the implementation of a linear scaling limiter when thenumerical solution leaves the set of realizable moments (that is, those momentsassociated with a positive underlying distribution). The difficulty for such alimiter lies in the computation of the intersection of a ray with the set ofrealizable moments. We avoid this computation by using quadrature to generate aconvex polytope which approximates this set. The half-space representation ofthis polytope is used to compute an approximation of the required intersectionstraightforwardly, and with this limiter in hand, the rest of the DG scheme isconstructed using standard techniques. We consider the resulting numericalscheme on a new manufactured solution and standard benchmark problems for bothtraditional M$_N$ models and the so-called mixed-moment models. Themanufactured solution allows us to observe the expected convergence rates andexplore the effects of the regularization in the optimization.