A blob method for diffusion

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.

Entropy solutions for stochastic porous media equations

“…Informally, localising the particle interactions, by taking V to a Dirac mass, leads to a stochastic partial differential equation (SPDE) of the type (1.1), albeit with space-time white noise. So far, this last step has rigorously been justified only in the deterministic case by Lions, Mas Gallic [LMG01] and Carrillo, Craig, Patacchini [CCP19].…”

Discretization of flux-limited gradient flows: $\Gamma $-convergence and numerical schemes

“…The proof is based on the Γ-convergence of the functional in (11) to the one in (7) without E(ρ n−1 ), which is made precise in Proposition 1 below. That Γ-limit would be fairly easy to obtain in the situation of regular cost functions, i.e., when C is a continuous and strictly convex function on all of R d .…”

“…This is extremely efficient in one space dimension [7,22,23], but becomes significantly more cumbersome -and difficult to analyze -in multiple dimensions [5,12,13,18]. Various alternatives to the Lagrangian approach are available, including finite volume methods [21], blob methods [11] etc.…”

Fisher information regularization schemes for Wasserstein gradient flows

“…, which substituting back into the constraint of (13) leads to the same flux for ρ(1, x) as in (15).…”

“…But they often suffer from stability constraints, due either to the degeneracy of the diffusion or the non-locality from the interaction potential, such as the mesa problem [51]. Another approach leverages structural similarities between (1) and equations from fluid dynamics to develop particle methods [9,12,15,36,53]. On one hand, particle methods naturally conserve mass and positivity, and they can also be designed to respect the underlying gradient flow structure of the equation so as to dissipate the energy along time.…”