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

In this paper, we consider stochastic dynamical systems on the sphere and the associated Fokker–Planck equations. A semi-Lagrangian method combined with a Finite Volume discretization of the sphere is presented to solve the Fokker–Planck equation. The method is applied to a typical problem in fiber dynamics and textile production. The numerical results are compared to explicit solutions and Monte-Carlo solutions.

[1]  Axel Klar,et al.  Hierarchy of mathematical models for production processes of technical textiles , 2009 .

[2]  Vicsek,et al.  Novel type of phase transition in a system of self-driven particles. , 1995, Physical review letters.

[3]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[4]  Axel Klar,et al.  A 3D MODEL FOR FIBER LAY-DOWN IN NONWOVEN PRODUCTION PROCESSES , 2012 .

[5]  Akio Arakawa,et al.  Integration of the Nondivergent Barotropic Vorticity Equation with AN Icosahedral-Hexagonal Grid for the SPHERE1 , 1968 .

[6]  Axel Klar,et al.  Hydrodynamic Limit of a Fokker--Planck Equation Describing Fiber Lay-Down Processes , 2007, SIAM J. Appl. Math..

[7]  Axel Klar,et al.  Ergodicity and Rate of Convergence for a Nonsectorial Fiber Lay-Down Process , 2008, SIAM J. Math. Anal..

[8]  Mladen Savov,et al.  (Non-)Ergodicity of a Degenerate Diffusion Modeling the Fiber Lay Down Process , 2013, SIAM J. Math. Anal..

[9]  Bernd Simeon,et al.  A space-time adaptive approach to orientation dynamics in particle laden flows , 2010, ICCS.

[10]  Pierre Degond,et al.  Continuum limit of self-driven particles with orientation interaction , 2007, 0710.0293.

[11]  Michael Manhart,et al.  On the numerical solution of a convection-diffusion equation for particle orientation dynamics on geodesic grids , 2012 .

[12]  P. Degond,et al.  A Hierarchy of Heuristic-Based Models of Crowd Dynamics , 2013, 1304.1927.

[13]  T. F. Russell,et al.  NUMERICAL METHODS FOR CONVECTION-DOMINATED DIFFUSION PROBLEMS BASED ON COMBINING THE METHOD OF CHARACTERISTICS WITH FINITE ELEMENT OR FINITE DIFFERENCE PROCEDURES* , 1982 .

[14]  Axel Klar,et al.  Geometry, mixing properties and hypocoercivity of a degenerate diffusion arising in technical textile industry , 2012, 1203.4502.

[15]  A. Kageyama,et al.  ``Yin-Yang grid'': An overset grid in spherical geometry , 2004, physics/0403123.

[16]  Daniel W. Stroock,et al.  On the growth of stochastic integrals , 1971 .

[17]  Chieh-Sen Huang,et al.  The modified method of characteristics with adjusted advection , 1999, Numerische Mathematik.

[18]  E. Wachspress,et al.  A Rational Finite Element Basis , 1975 .

[19]  Axel Klar,et al.  A Stochastic Model and Associated Fokker-Planck Equation for the Fiber Lay-Down Process in Nonwoven Production Processes , 2007, SIAM J. Appl. Math..

[20]  Mohammed Seaïd,et al.  On the Quasi-monotone Modified Method of Characteristics for Transport-diffusion Problems with Reactive Sources , 2001 .

[21]  Chris Hill,et al.  Implementation of an Atmosphere-Ocean General Circulation Model on the Expanded Spherical Cube , 2004 .

[22]  B. Øksendal.  Stochastic Differential Equations , 1985 .

[23]  Axel Klar,et al.  A Semi-Lagrangian Method for a Fokker-Planck Equation Describing Fiber Dynamics , 2009, J. Sci. Comput..