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 nonoverlapping 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.with superlinear growth at infinity and H(0) = 0. Furthermore, h(λ) := λ d H λ −d is convex and non-increasing on (0, ∞).

Entropy Dissipation Semi-Discretization Schemes for Fokker–Planck Equations

“…which is different from the commonly adopted centered difference. We call (9) the "Log-Laplacian." We observe that the Log-Laplacian plays a crucial role in the spatial discretization.…”

“…In the literature, people have studied 2-Wasserstein metric and Fokker-Planck equations in discrete settings for a long time [7,8,9,15,18,20]. Maas [18] and Mielke [20] introduce a different discrete 2-Wasserstein metric.…”

“…Our scheme shows exponential convergence for all linear and nonlinear cases. Carrillo, Chertock, Huang, Wolansky [8,9] have recently designed several algorithms based on entropy dissipation viewpoint. Particularly, the first order scheme designed in [8] shares some similarities with (3) for a lattice graph.…”

Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms

“…Many of the Lagrangian methods are based on the "discretize-then-minimize" strategy, see for example [56,28,45,7,21,41]. However, we follow closely the advantage of the 1D formulation in terms of the pseudo-inverse function, see also [34,33,10], which corresponds to a "minimize-then-discretize" strategy.…”

“…The variational formulation provides an underlying structure for the construction of numerical methods, which have inherent advantages such as build-in positivity and free-energy decrease. There has been an increasing interest in the development of such methods in the last years, for example variational Lagrangian schemes such as [34,33,56,28,10,45,21,41] or finite volume schemes as in [23]. The common challenge of all methods is the high computational complexity, often restricting them to spatial dimension one.…”