[Paper Review] A variational finite volume scheme for Wasserstein gradient flows

Clément Cancès, Thomas Gallouët+1 more|arXiv (Cornell University)|Jul 18, 2019

Geometric Analysis and Curvature FlowsMathematics68 references27 citations

TL;DR

This paper presents a novel variational finite volume scheme for solving Wasserstein gradient flows by combining time discretization via the JKO scheme with a space-discretized upstream mobility two-point flux approximation. The method preserves the variational structure of the continuous problem, guarantees non-negativity and energy decay, and is proven convergent for the linear Fokker-Planck equation. It achieves first-order accuracy in both time and space and is robust across diverse energies and initial profiles.

ABSTRACT

We propose a variational finite volume scheme to approximate the solutions to Wasserstein gradient flows. The time discretization is based on an implicit linearization of the Wasserstein distance expressed thanks to Benamou-Brenier formula, whereas space discretization relies on upstream mobility two-point flux approximation finite volumes. Our scheme is based on a first discretize then optimize approach in order to preserve the variational structure of the continuous model at the discrete level. Our scheme can be applied to a wide range of energies, guarantees non-negativity of the discrete solutions as well as decay of the energy. We show that our scheme admits a unique solution whatever the convex energy involved in the continuous problem, and we prove its convergence in the case of the linear Fokker-Planck equation with positive initial density. Numerical illustrations show that it is first order accurate in both time and space, and robust with respect to both the energy and the initial profile.

Motivation & Objective

• To develop a robust, structure-preserving numerical scheme for Wasserstein gradient flows arising in PDEs like the Fokker-Planck and porous medium equations.
• To preserve the variational structure of the continuous model at the discrete level through a first-discretize-then-optimize approach.
• To ensure discrete non-negativity and energy decay for a wide class of convex energies.
• To prove convergence of the scheme for the linear Fokker-Planck equation with positive initial density.
• To demonstrate robustness and first-order accuracy in numerical experiments across diverse physical scenarios.

Proposed method

• Time discretization is based on the JKO semi-discretization, reformulating the problem as a minimization involving the Wasserstein distance via the Benamou-Brenier dynamic formulation.
• Space discretization employs a finite volume method with upstream mobility two-point flux approximation to handle convection-dominated flows.
• The scheme is derived from a primal-dual saddle-point problem involving a forward continuity equation and a backward Hamilton-Jacobi equation, leading to a mean-field game system.
• The discrete energy is minimized subject to mass conservation, and the dual problem is solved via a fixed-point iteration on the dual variable (Hamilton-Jacobi potential).
• The method ensures unique solvability for any convex energy and maintains the variational structure via a discrete version of the Benamou-Brenier formula.
• The scheme is implemented using a piecewise constant finite volume formulation on unstructured meshes, with the dual variable updated iteratively to enforce optimality.

Experimental results

Research questions

• RQ1 Can a finite volume scheme be constructed that preserves the variational structure of Wasserstein gradient flows at the discrete level?
• RQ2 Does the proposed scheme guarantee discrete non-negativity and energy decay for general convex energies?
• RQ3 Is the scheme convergent for the linear Fokker-Planck equation with positive initial density?
• RQ4 What is the convergence rate of the scheme in time and space, and how robust is it to different initial profiles and energy functionals?
• RQ5 Can the scheme be extended to systems of equations, such as those modeling salinity intrusion in aquifers?

Key findings

• The scheme is proven to have a unique solution for any convex energy functional, ensuring numerical stability regardless of the energy type.
• For the linear Fokker-Planck equation, the scheme converges to the weak solution of the continuous problem as the time step and mesh size tend to zero.
• Numerical results show first-order convergence in both time and space, with convergence profiles improving on refined meshes and smaller time steps.
• The scheme dissipates energy faster than standard finite volume schemes, which is consistent with its design to maximize energy decay at each step.
• The energy dissipation of the scheme approaches the true dissipation of the continuous system as the mesh is refined and the time step is reduced.
• The scheme successfully simulates complex physical phenomena such as the porous medium equation and salinity intrusion, even when theoretical convergence is not proven for such systems.