First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions

Abstract

Mixed-moment models, introduced in [14,29] for one space dimension, are a modification of the method of moments applied to a (linear) kinetic equation, by choosing mixtures of different partial moments. They are well-suited to handle such equations where collisions of particles are modelled with a Laplace-Beltrami operator. We generalize the concept of mixed moments to two dimension. The resulting hyperbolic system of equations has desirable properties, removing some drawbacks of the well-known M 1 model. We furthermore provide a realizability theory for a first-order system of mixed moments by linking it to the corresponding quarter-moment theory. Additionally, we derive a type of Kershaw closures for mixed-and quarter-moment models, giving an efficient closure (compared to minimum-entropy models). The derived closures are investigated for different benchmark problems.

Partial-moment minimum-entropy models for kinetic chemotaxis equations in one and two dimensions

“…For example, the second-order M 2 model already provides reasonably better results in case of the two-dimensional twospikes example (see e.g. [38] for a similar test case). However, although this model has less degrees of freedoms (six equations in total while the QM 1 model has twelve) it is much more expensive.…”

“…For example, the first moment has to be located within the corresponding quarter sphere, i.e. u ±± ∈ v(V ±± ) [16,34,38]. Equations for the moments u can be obtained by testing (2.1) with 1 and v, and integrating over the four quarter spheres V ±± :…”

“…[18]. Similar tests have been used in [26,38] to investigate the M 1 and QM 1 model in case of fibre-laydown and radiative transfer equations. In the following we compare the results of those four computations: The simulation obtained by computing both spikes separately and then superposing the results acts as our reference and is pictured in Figure 9.…”

Second-order mixed-moment model with differentiable ansatz function in slab geometry

“…Furthermore, the DMM N should be investigated in higher dimensions, especially in the context of the Fokker-Planck operator. The results in [34,37] indicate that mixed moments are hardly applicable in this framework due to the difficulty in the discretization of the Laplace-Beltrami operator. This should be avoidable using the differentiable basis functions.…”

Kershaw closures for linear transport equations in slab geometry II: High-order realizability-preserving discontinuous-Galerkin schemes

“…Finally, the concepts have to be lifted to higher dimensions. While fully three-dimensional first-order variants of Kershaw closures exist [17,35], no higher-order models or a completely closed theory is available. With this, generalizing the presented scheme is in principle possible and it can be expected that similar efficiency results hold true.…”