A Fully Discrete Variational Scheme for Solving Nonlinear Fokker-Planck Equations in Multiple Space Dimensions

We introduce a novel spatio-temporal discretization for nonlinear Fokker-Planck equations on the multi-dimensional unit cube. This discretization is based on two structural properties of these equations: the first is the representation as a gradient flow of an entropy functional in the $L^2$-Wasserstein metric, the second is the Lagrangian nature, meaning that solutions can be written as the push forward transformation of the initial density under suitable flow maps. The resulting numerical scheme is entropy diminishing and mass conserving. Further, the scheme is weakly stable, which allows us to prove convergence under certain regularity assumptions. Finally, we present results from numerical experiments in space dimension $d=2$.

[1]  H. Hadwiger,et al.  Brunn-Minkowskischer Satz und Isoperimetrie , 1956 .

[2]  Quentin Mérigot,et al.  Discretization of functionals involving the Monge–Ampère operator , 2014, Numerische Mathematik.

[3]  José A. Carrillo,et al.  Convergence of the Mass-Transport Steepest Descent Scheme for the Subcritical Patlak-Keller-Segel Model , 2008, SIAM J. Numer. Anal..

[4]  Giovanni Russo,et al.  Deterministic diffusion of particles , 1990 .

[5]  Michael Westdickenberg,et al.  A time discretization for the pressureless gas dynamics equations , 2014 .

[6]  M. Agueh,et al.  One-Dimensional Numerical Algorithms for Gradient Flows in the p-Wasserstein Spaces , 2013 .

[7]  J. Vázquez.  An Introduction to the Mathematical Theory of the Porous Medium Equation , 1992 .

[8]  A Wasserstein approach to the numerical solution of the one-dimensional Cahn-Hilliard equation , 2010 .

[9]  Daniel Matthes,et al.  A Convergent Lagrangian Discretization for a Nonlinear Fourth-Order Equation , 2017, Found. Comput. Math..

[10]  J. Carrillo,et al.  On the asymptotic behavior of the gradient flow of a polyconvex functional , 2009 .

[11]  D. Matthes,et al.  Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation , 2013, 1301.0747.

[12]  E. Tadmor.  Spectral methods for time dependent problems , 1990 .

[13]  Robert D. Russell,et al.  Self–similar numerical solutions of the porous–medium equation using moving mesh methods , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[14]  L. Ambrosio,et al.  Stability of flows associated to gradient vector fields and convergence of iterated transport maps , 2006, manuscripta mathematica.

[15]  Marie-Therese Wolfram,et al.  Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms , 2016, J. Comput. Phys..

[16]  J. A. Carrillo,et al.  Numerical Simulation of Diffusive and Aggregation Phenomena in Nonlinear Continuity Equations by Evolving Diffeomorphisms , 2009, SIAM J. Sci. Comput..

[17]  L. C. Evans,et al.  Diffeomorphisms and Nonlinear Heat Flows , 2005, SIAM Journal on Mathematical Analysis.

[18]  F. Otto.  THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION , 2001 .

[19]  Daniel Matthes,et al.  A gradient flow scheme for nonlinear fourth order equations , 2010 .

[20]  E. Socolovsky,et al.  A numerical procedure for the porous media equation , 1985 .

[21]  Michael Westdickenberg,et al.  VARIATIONAL PARTICLE SCHEMES FOR THE POROUS MEDIUM EQUATION AND FOR THE SYSTEM OF ISENTROPIC EULER EQUATIONS , 2008, 0807.3573.

[22]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[23]  Laurent Gosse,et al.  Identification of Asymptotic Decay to Self-Similarity for One-Dimensional Filtration Equations , 2006, SIAM J. Numer. Anal..

[24]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .