A blob method for diffusion

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 we 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, 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 energies converge to solutions of the porous medium equation. As a corollary, we obtain convergence of our numerical blob method, again under sufficient regularity assumptions. 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

Numerical validation of an adaptive model for the determination of nonlinear-flow regions in highly heterogeneous porous media

An adaptive model for the description of flows in highly heterogeneous porous media is developed in~\cite{FP21,FP23}. There, depending on the magnitude of the fluid's velocity, the constitutive law linking velocity and pressure gradient is selected between two possible options, one better adapted to slow motion and the other to fast motion. We propose here to validate further this adaptive approach by means of more extensive numerical experiments, including a three-dimensional case, as well as to use such approach to determine a partition of the domain into slow- and fast-flow regions

Nonlocal-interaction equation on graphs: gradient flow structure and continuum limit

We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou-Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of "vertices" is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL$^2$IE). We develop the existence theory for the solutions of the NL$^2$IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$^2$IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.Comment: 46 pages. Minor revision with improved presentation and fixed typo

Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL2IE). We develop the existence theory for the solutions of the NL2IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL2IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result