First-order continuous- and discontinuous-Galerkin moment models for a linear kinetic equation: realizability-preserving splitting scheme and numerical analysis

Abstract

We derive a second-order realizability-preserving scheme for moment models for linear kinetic equations. We apply this scheme to the first-order continuous (HFM n ) and discontinuous (PMM n ) models in slab and three-dimensional geometry derived in [54] as well as the classical full-moment M N models. We provide extensive numerical analysis as well as our code to show that the new class of models can compete or even outperform the full-moment models in reasonable test cases.

The second-order formulation of the $P_N$ equations with Marshak boundary conditions

“…This test case demonstrates that our method can deal with heterogeneous coefficients in 2D. It is based on the shadow test [39,50] which represents a particle stream that is partially blocked by an absorber, resulting in a shadowed region behind the absorber. The setup for this test case in 2D is given in Table 9, based on the auxiliary functions in Equation (31).…”

“…In recent years many modifications to this closure have been suggested, including the positive P N (PP N ), filtered P N (FP N ) and diffusive-corrected P N (D N ) [38], curing some of the disadvantages of the original P N method while increasing the complexity of the system at the price of higher computational costs. We also want to note that the choices of other closures and angular bases are possible, e.g., minimum entropy [3,31,39,[39][40][41][42][43][44][45][46][47][48][49], partial and mixed moments [50][51][52][53][54][55] or Kershaw closures [56][57][58][59].…”

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

“…Let α one ∈ R n be a vector such that α T one b ≡ 1 (such a vector exists for all regarded basis [44]). Using (2.4) and (2.5), from u we can get the local particle density as…”

“…Checking realizability is much easier when using piecewise linear bases instead of the standard polynomial basis on the whole velocity space [13,14,35,42,[44][45][46]. In addition, the computational cost is significantly lower for these models.…”

“…In addition, the computational cost is significantly lower for these models. However, solving the optimization problems is still costly compared to linear models and maintaining realizability still requires additional limiters [44].…”