Numerical Study of a Particle Method for Gradient Flows

Abstract

We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting scheme preserves the gradient flow structure at the particle level, and enables us to obtain a gradient descent formulation after time discretisation. We give several simulations to illustrate the validity of this method, as well as a detailed study of one-dimensional aggregation-diffusion equations.

27 pages, 21 figures

References in corpus (6)

• A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure
• Convergence of a particle method for diffusive gradient flows in one dimension
• Discretization of functionals involving the Monge-Ampère operator
• A fully discrete variational scheme for solving nonlinear Fokker-Planck equations in higher space dimensions
• Convergence of a fully discrete variational scheme for a thin-film equation
• Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up

Cited by in corpus (14)

• A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials
• Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms
• On Lagrangian schemes for porous medium type generalized diffusion equations: a discrete energetic variational approach
• A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes
• Langevin Monte Carlo and JKO splitting
• Primal dual methods for Wasserstein gradient flows
• On gradient flow and entropy solutions for nonlocal transport equations with nonlinear mobility
• Convergence of a particle method for diffusive gradient flows in one dimension
• Entropy dissipation semi-discretization schemes for Fokker-Planck equations
• A Variational Lagrangian Scheme for a Phase Field Model: A Discrete Energetic Variational Approach
• Dataset Dynamics via Gradient Flows in Probability Space
• Structure-Preserving Numerical Methods for Nonlinear Fokker--Planck Equations with Nonlocal Interactions by an Energetic Variational Approach
• Gradient flows of interacting Laguerre cells as discrete porous media flows
• On the gradient flow structure of the isotropic Landau equation