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

Mixed-moment minimum-entropy models (${\rm{M}}{{\rm{M}}_N}$ models) are known to overcome the zero net-flux problem of full-moment minimum entropy ${{\rm{M}}_N}$ models but lack regularity. We study differentiable mixed-moment models (full zeroth and first moment, half higher moments, called ${\rm{DM}}{{\rm{M}}_N}$ models) for a Fokker-Planck equation in one space dimension. Realizability theory for these modification of mixed moments is derived for second order. Numerical tests are performed with a kinetic first-order finite volume scheme and compared with ${{\rm{M}}_N}$, classical ${\rm{M}}{{\rm{M}}_N}$ and a ${{\rm{P}}_N}$ reference scheme.

keywords: Moment models minimum entropy Fokker-Planck equation realizability mixed moments
Approximate models for stochastic dynamic systems with velocities on the sphere and associated Fokker--Planck equations
Axel Klar Florian Schneider Oliver Tse
We consider stochastic dynamic systems with state space $\mathbb{R}^n \times \mathbb{S}^{n-1}$ and associated Fokker--Planck equations. Such systems are used to model, for example, fiber dynamics or swarming and pedestrian dynamics with constant individual speed of propagation. Approximate equations, like linear and nonlinear (maximum entropy) moment approximations and linear and nonlinear diffusion approximations are investigated. These approximations are compared to the underlying Fokker--Planck equation with respect to quality measures like the decay rates to equilibrium. The results clearly show the superiority of the maximum entropy approach for this application compared to the simpler linear and diffusion approximations.
keywords: maximum entropy Fokker--Planck equations stochastic dynamic system exponential decay to equilibrium.
First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions
Florian Schneider Andreas Roth Jochen Kall

Mixed-moment models, introduced in [8,44] 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 equations where collisions of particles are modelled with a Laplace-Beltrami operator. We generalize the concept of mixed moments to two dimensions. In the context of minimum-entropy models, the resulting hyperbolic system of equations has desirable properties (entropy-diminishing, bounded eigenvalues), removing some drawbacks of the well-known M1 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.

keywords: Radiation transport moment models realizability mixed moments Laplace-Beltrami operator
A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry
Florian Schneider Jochen Kall Graham Alldredge
We develop a high-order kinetic scheme for entropy-based moment models of a one-dimensional linear kinetic equation in slab geometry. High-order spatial reconstructions are achieved using the weighted essentially non-oscillatory (WENO) method. For time integration we use multi-step Runge-Kutta methods which are strong stability preserving and whose stages and steps can be written as convex combinations of forward Euler steps. We show that the moment vectors stay in the realizable set using these time integrators along with a maximum principle-based kinetic-level limiter, which simultaneously dampens spurious oscillations in the numerical solutions. We present numerical results both on a manufactured solution, where we perform convergence tests showing our scheme has the expected order up to the numerical noise from the optimization routine, as well as on two standard benchmark problems, where we show some of the advantages of high-order solutions and the role of the key parameter in the limiter.
keywords: moment models realizability-preserving kinetic scheme Radiation transport high order WENO. realizability

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