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

Abstract

We implement a high-order numerical scheme for the entropy-based moment closure, the so-called M${}_N$ model, for linear kinetic equations in slab geometry. A discontinuous Galerkin (DG) scheme in space along with a strong-stability preserving Runge-Kutta time integrator is a natural choice to achieve a third-order scheme, but so far, the challenge for such a scheme in this context is the implementation of a linear scaling limiter when the numerical solution leaves the set of realizable moments (that is, those moments associated with a positive underlying distribution). The difficulty for such a limiter lies in the computation of the intersection of a ray with the set of realizable moments. We avoid this computation by using quadrature to generate a convex polytope which approximates this set. The half-space representation of this polytope is used to compute an approximation of the required intersection straightforwardly, and with this limiter in hand, the rest of the DG scheme is constructed using standard techniques. We consider the resulting numerical scheme on a new manufactured solution and standard benchmark problems for both traditional M${}_N$ models and the so-called mixed-moment models. The manufactured solution allows us to observe the expected convergence rates and explore the effects of the regularization in the optimization.

Cited by in corpus (8)

• A hyperbolicity-preserving stochastic Galerkin approximation for uncertain hyperbolic systems of equations
• A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry
• Kershaw closures for linear transport equations in slab geometry I: model derivation
• Kershaw closures for linear transport equations in slab geometry II: high-order realizability-preserving discontinuous-Galerkin schemes
• On the convergence of the regularized entropy-based moment method for kinetic equations
• First-order quarter- and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions
• First-order continuous- and discontinuous-Galerkin moment models for a linear kinetic equation: model derivation and realizability theory
• Implicit-explicit, realizability-preserving first-order scheme for moment models with Lipschitz-continuous source terms