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

Abstract

We introduce a novel spatio-temporal discretization for nonlinear Fokker--Planck equations on the multidimensional 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, we are able to prove consistency in the sense that if the discrete solutions are regular independently of the mesh size, then they converge to a classical solution. Finally, we present results from numerical experiments in space dimension d=2.

Keywords

  1. Fokker--Planck equation
  2. gradient flow
  3. Lagrangian discretization
  4. Wasserstein metric
  5. structure preservation

MSC codes

  1. Primary
  2. 65M12; Secondary
  3. 35K59
  4. 65K10

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Information & Authors

Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 419 - 443
ISSN (online): 1095-7170

History

Submitted: 13 January 2016
Accepted: 15 November 2016
Published online: 28 February 2017

Keywords

  1. Fokker--Planck equation
  2. gradient flow
  3. Lagrangian discretization
  4. Wasserstein metric
  5. structure preservation

MSC codes

  1. Primary
  2. 65M12; Secondary
  3. 35K59
  4. 65K10

Authors

Affiliations

Funding Information

Deutsche Forschungsgemeinschaft http://dx.doi.org/10.13039/501100001659 : Collaborative Research Center TRR 109 Discretization in Geometry and Dynamics

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