Second-order mixed-moment model with differentiable ansatz function in slab geometry
Florian Schneider
Kinetic & Related Models 2018, 11(5): 1255-1276 doi: 10.3934/krm.2018049

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
Kinetic & Related Models 2014, 7(3): 509-529 doi: 10.3934/krm.2014.7.509
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
Kinetic & Related Models 2017, 10(4): 1127-1161 doi: 10.3934/krm.2017044

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

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