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 …

Given a large ensemble of interacting particles, driven by nonlocal interactions and localized
repulsion, the mean-field limit leads to a class of nonlocal, nonlinear partial differential …

Combining the classical theory of optimal transport with modern operator splitting
techniques, we develop a new numerical method for nonlinear, nonlocal partial differential …

Solutions to many partial differential equations satisfy certain bounds or constraints. For
example, the density and pressure are positive for equations of fluid dynamics, and in the …

We develop a set of numerical schemes for the Poisson–Nernst–Planck equations. We
prove that our schemes are mass conservative, uniquely solvable and keep positivity …

We propose a new Lagrange multiplier approach to construct positivity preserving schemes
for parabolic type equations. The new approach introduces a space–time Lagrange …

We propose numerical schemes for a class of Keller--Segel equations. The discretization is
based on the gradient flow structure. The resulting first-order scheme is mass conservative …

We propose a new method to construct high-order, linear, positivity/bound preserving and
unconditionally energy stable schemes for general dissipative systems whose solutions are …

We propose a variational scheme for computing Wasserstein gradient flows. The scheme
builds upon the Jordan–Kinderlehrer–Otto framework with the Benamou-Brenier's dynamic …

We construct two classes of time discretization schemes for fourth order nonlinear equations
by combining a function transform approach with the scalar auxiliary variable (SAV) …