A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension
This work provides a practical high-order numerical method for entropy-based moment closures in kinetic equations, addressing a key computational bottleneck for researchers in computational physics.
The authors developed a high-order discontinuous Galerkin scheme for entropy-based moment closures (M_N models) for linear kinetic equations, overcoming the realizability-preserving challenge by approximating the set of realizable moments with a convex polytope via quadrature. The scheme achieves third-order accuracy and is validated on manufactured solutions and benchmark problems.
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.