A Semi-Lagrangian Method for 3-D Fokker Planck Equations for Stochastic Dynamical Systems on the Sphere

Röth, Andreas; Klar, Axel; Simeon, Bernd; Zharovsky, Evgeniy

doi:10.1007/s10915-014-9835-z

Cited by 4 publications

(7 citation statements)

References 17 publications

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“…The approximation for f (t n+ 1 2 , x, τ ij ) is similar to the Lax-Wendroff numerical flux function, except that we evaluate F (x, τ ) · e(τ ) at the edge midpoint τ ij , and not at the cell midpoints τ i and τ j , since the grid structure only provides normal vectors at the cell interfaces. The value for ∇ τ f (t n+ 1 2 , x, τ ij ) · e(τ ij ) is obtained by a finite difference approximation on the connecting circle arc of the cell midpoints τ i and τ j , see [27]. Although the method is not a second order method, the numerical results are close to those of a second order method, see the discussion below and Figure 5.2 for the convergence rates of the splitting scheme.…”

Section: Mean Field Equationmentioning

confidence: 78%

“…Mean field equation. The numerical methods used here are an advancement of the schemes used in [9,27,28]. We apply a second order Strang splitting [10] to (3.1) to obtain subproblems on spatial and velocity domain.…”

Section: Numerical Methods and Resultsmentioning

confidence: 99%

“…Firstly, we discuss the solution of (5.3) and (5.5) with a finite volume discretization of the velocity space based on a geodesic grid (cf. [27,29]). The geodesic grid consists of spherical triangles, which are represented with vertices and normals in R 3 , neighbor relations and correct distance and surface measures.…”

Section: Numerical Methods and Resultsmentioning

confidence: 99%

“…The analysis of the limit is based on the work in [5,12,15,16,17,25,31]. For numerical methods for mean-field type equations we refer to [2,26,27,30].…”

mentioning

confidence: 99%

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A Retarded Mean-Field Approach for Interacting Fiber Structures

Borsche¹,

Klar²,

Nessler³

et al. 2017

Multiscale Model. Simul.

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Abstract. We consider an interacting system of one-dimensional structures modelling fibers with fiber-fiber interaction in a fiber lay-down process. The resulting microscopic system is investigated by looking at different asymptotic limits of the corresponding stochastic model. Equations arising from mean-field and diffusion limits are considered. Furthermore, numerical methods for the stochastic system and its mean-field counterpart are discussed. A numerical comparison of solutions corresponding to the different scales (microscopic, mesoscopic and macroscopic) is included.Keywords interacting stochastic particles, fibers, mean-field equations, retarded potential, delay equations.2010 AMS Subject Classification: 92D50, 35B40, 82C22, 92C151. Introduction. One-dimensional structures appear in various context of industrial applications. They are used, for example, in the modelling of polymers in suspensions, composite materials, nanostructures, fiber dynamics in turbulent flows and, in particular, fiber lay-down in technical processes of non-woven materials. Furthermore, such structures have been modelled on different levels of description involving different scales. Besides microscopic models, mesoscopic kinetic or Fokker-Planck equations have been widely used for a statistical description of the fiber or polymer distributions. We refer to [1,2] and [18,21] for concrete examples in the industry. In this article, we consider non-woven materials, which are webs of long flexible fibers. Production processes and models corresponding to the lay-down of such fibers have been intensively investigated. See [4] and the above cited references.In the above mentioned investigations, the one-dimensional structures (fibers) under consideration were assumed to be mutually independent, which clearly does not represent reality. Therefore, the present work aims at including fiber-fiber interaction, thereby describing the size of each fiber and, simultaneously, the absence of intersection among fibers. This is achieved by simply including the interaction of structures into a well investigated model described in non-woven production processes. Taking into account the interaction of the structures on the microscopic level leads to coupled systems of stochastic differential equations. Its statistical description should also take into account the interactions and will consequently no longer be based on the classical Fokker-Planck model.The new model makes use of a microscopic systems of retarded stochastic differential equations, and its mesoscopic description is obtained via formal mean-field procedures. The mean-field limit is described by a McKean-Vlasov type equation with a delay term. We perform an analytical investigation of the mean field limit, as well as a numerical comparison of microscopic, mean field and macroscopic equations. The analysis of the limit is based on the work in [5,12,15,16,17,25,31]. For numerical methods for mean-field type equations we refer to [2,26,27,30].The paper is organized as follows: starting from a...

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Section: Mean Field Equationmentioning

confidence: 78%

Section: Numerical Methods and Resultsmentioning

confidence: 99%

Section: Numerical Methods and Resultsmentioning

confidence: 99%

“…The analysis of the limit is based on the work in [5,12,15,16,17,25,31]. For numerical methods for mean-field type equations we refer to [2,26,27,30].…”

mentioning

confidence: 99%

See 2 more Smart Citations

A Retarded Mean-Field Approach for Interacting Fiber Structures

Borsche¹,

Klar²,

Nessler³

et al. 2017

Multiscale Model. Simul.

Self Cite

View full text Add to dashboard Cite

show abstract

“…For the advection problem (3.2), our procedure can be easily extended to 2D by extending the above described vectors in an appropriate way. For the discretization in velocity, we apply a finite volume scheme [46] of second order with a Lax Wendroff type approximation [54] for the advection, which is an enhancement of the methods used in [56,64]: Denote the midpoints of rectangular cells T k for (2.3) by v k . The boundary between cells k and l is called T kl with the normal e(v) and the boundary midpoint v kl .…”

Section: Numerical Schemementioning

confidence: 99%

Single to double mill small noise transition via semi-Lagrangian finite volume methods

Carrillo

Klar

Röth

2016

Communications in Mathematical Sciences

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We show that double mills are more stable than single mills under stochastic perturbations in swarming dynamic models with basic attraction-repulsion mechanisms. In order to analyse accurately this fact, we will present a numerical technique for solving kinetic mean field equations for swarming dynamics. Numerical solutions of these equations for different sets of parameters will be presented and compared to microscopic and macroscopic results. As a consequence, we numerically observe a phase transition diagram in term of the stochastic noise going from single to double mill for small stochasticity fading gradually to disordered states when the noise strength gets larger. This bifurcation diagram at the inhomogeneous kinetic level is shown by carefully computing the distribution function in velocity space.

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

Schneider¹,

Röth²,

Kall³

2017

Kinetic &Amp; Related Models

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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.

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A Semi-Lagrangian Method for 3-D Fokker Planck Equations for Stochastic Dynamical Systems on the Sphere

Cited by 4 publications

References 17 publications

A Retarded Mean-Field Approach for Interacting Fiber Structures

A Retarded Mean-Field Approach for Interacting Fiber Structures

Single to double mill small noise transition via semi-Lagrangian finite volume methods

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

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