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Discretizing the Fokker-Planck equation with second-order accuracy: a dissipation driven approach

Clément Cancès, Léonard Monsaingeon, Andrea Natale

Abstract

We propose a fully discrete finite volume scheme for the standard Fokker-Planck equation. The space discretization relies on the well-known square-root approximation, which falls into the framework of two-point flux approximations. Our time discretization is novel and relies on a tailored nonlinear mid-point rule, designed to accurately capture the dissipative structure of the model. We establish well-posedness for the scheme, positivity of the solutions, as well as a fully discrete energy-dissipation inequality mimicking the continuous one. We then prove the rigorous convergence of the scheme under mildly restrictive conditions on the unstructured grids, which can be easily satisfied in practice. Numerical simulations show that our scheme is second order accurate both in time and space, and that one can solve the discrete nonlinear systems arising at each time step using Newton's method with low computational cost.

Discretizing the Fokker-Planck equation with second-order accuracy: a dissipation driven approach

Abstract

We propose a fully discrete finite volume scheme for the standard Fokker-Planck equation. The space discretization relies on the well-known square-root approximation, which falls into the framework of two-point flux approximations. Our time discretization is novel and relies on a tailored nonlinear mid-point rule, designed to accurately capture the dissipative structure of the model. We establish well-posedness for the scheme, positivity of the solutions, as well as a fully discrete energy-dissipation inequality mimicking the continuous one. We then prove the rigorous convergence of the scheme under mildly restrictive conditions on the unstructured grids, which can be easily satisfied in practice. Numerical simulations show that our scheme is second order accurate both in time and space, and that one can solve the discrete nonlinear systems arising at each time step using Newton's method with low computational cost.
Paper Structure (24 sections, 13 theorems, 216 equations, 6 figures, 4 tables)

This paper contains 24 sections, 13 theorems, 216 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Fokker-Planck equation and Wasserstein gradient flows
  3. Our contribution and organization of the paper
  4. A space-time discretization for the Fokker-Planck equation
  5. Finite volume discretization
  6. Numerical scheme
  7. Discrete well-posedness and dissipative structure
  8. Existence and uniqueness of solutions
  9. Discrete energy dissipation equality
  10. Convergence via the energy dissipation equality
  11. Assumptions on the sequence of meshes
  12. Compactness and limit densities
  13. Strong convergence of the approximate densities
  14. Asymptotic lower bound for the discrete Fisher information
  15. Asymptotic lower bound for the discrete Benamou-Brenier functional
  16. ...and 9 more sections

Key Result

Lemma 2.1

There holds with equality if $c\geq e^{-1}a$, and moreover

Figures (6)

  • Figure 1: A graphical representation of the functions $f$, $g$, $\Theta(a,\cdot)$ and $\Xi(a,\cdot)$.
  • Figure 2: Illustration of the refinement patterns used in the numerical tests; \ref{['fig:sfig1']} is the base mesh used for both refinement patterns, while \ref{['fig:sfig2']} and \ref{['fig:sfig3']} are the second meshes for the two different refinement patterns used in the tests.
  • Figure 3: Error in the energy balance $\Delta$, for $\delta=0$. The labels refer to different meshes and time steps corresponding to the row numbers of Table \ref{['tab:delta0sub']} (left, refinement by subdivision) and \ref{['tab:delta0rep']} (right, refinement by repetition)
  • Figure 4: Dissipation rates $\mathcal{D}_\psi(\boldsymbol{\theta}^{n+1/2})$ (a), and $\mathcal{D}_{\psi^*}(\boldsymbol{\theta}^{n+1/2})$ (b), for $\delta=0$ and using the finest mesh and time step, corresponding to the last row of Table \ref{['tab:delta0sub']} (left, refinement by subdivision) and \ref{['tab:delta0rep']} (right, refinement by repetition)
  • Figure 5: Density evolution for the second test case (note that the color scale is renormalized in each picture).
  • ...and 1 more figures

Theorems & Definitions (31)

  • Definition 1
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 21 more