Abstract
As a counterpoint to classical stochastic particle methods for diffusion, we develop a deterministic particle method for linear and nonlinear diffusion. At first glance, deterministic particle methods are incompatible with diffusive partial differential equations since initial data given by sums of Dirac masses would be smoothed instantaneously: particles do not remain particles. Inspired by classical vortex blob methods, we introduce a nonlocal regularization of our velocity field that ensures particles do remain particles and apply this to develop a numerical blob method for a range of diffusive partial differential equations of Wasserstein gradient flow type, including the heat equation, the porous medium equation, the Fokker–Planck equation, and the Keller–Segel equation and its variants. Our choice of regularization is guided by the Wasserstein gradient flow structure, and the corresponding energy has a novel form, combining aspects of the well-known interaction and potential energies. In the presence of a confining drift or interaction potential, we prove that minimizers of the regularized energy exist and, as the regularization is removed, converge to the minimizers of the unregularized energy. We then restrict our attention to nonlinear diffusion of porous medium type with at least quadratic exponent. Under sufficient regularity assumptions, we prove that gradient flows of the regularized porous medium energies converge to solutions of the porous medium equation. As a corollary, we obtain convergence of our numerical blob method. We conclude by considering a range of numerical examples to demonstrate our method’s rate of convergence to exact solutions and to illustrate key qualitative properties preserved by the method, including asymptotic behavior of the Fokker–Planck equation and critical mass of the two-dimensional Keller–Segel equation.
Highlights
For a range of partial differential equations, from the heat and porous medium equations to the Fokker–Planck and Keller–Segel equations, solutions can be characterized as gradient flows with respect to the quadratic Wasserstein distance
We develop a deterministic particle method for Wasserstein gradient flows
The goal of the present paper is to introduce a new deterministic particle method for equations of the form (1), with linear and nonlinear diffusion (m ≥ 1), that respects the problem’s underlying gradient flow structure and naturally extends to all dimensions
Summary
For a range of partial differential equations, from the heat and porous medium equations to the Fokker–Planck and Keller–Segel equations, solutions can be characterized as gradient flows with respect to the quadratic Wasserstein distance. They naturally preserve stationary states, since dissipation of the free energy provides inherent stability, and often capture the rate of asymptotic decay Another common strategy for preserving the gradient flow structure at the discrete level is to leverage the discrete-time variational scheme introduced by Jordan et al [55]. A natural way to circumvent this difficulty, at least in the case of linear diffusion (m = 1), is to consider a stochastic particle method, in which the particles evolve via Brownian motion Such approaches were originally developed in the classical fluids case [33], and several recent works have considered analogous methods for equations of Wasserstein gradient flow type, including the Keller–Segel equation [50, 52,53,62]. The main practical disadvantage of these stochastic methods is that their results must be averaged over a large number of runs to compensate for the inherent randomness
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