An optimal-transport finite-particle method for driven mass diffusion

TL;DR

This work develops a finite-particle method for mass diffusion that integrates a geometrically exact advection step, Wasserstein gradient-flow dynamics, and KL entropy regularization within a framework of mixed boundary conditions. A predictor-corrector scheme with an adsorption/depletion boundary layer enforces Dirichlet and Neumann conditions while particles are represented as Gaussian blobs to mollify density and enable a tractable energy-dissipation formulation. The authors demonstrate the approach on three test problems—sphere Dirichlet filling, mass storage in a mixed-boundary box, and pipe flow with inlet/outlet fluxes—observing convergence in the sense of measures and highlighting practical considerations for parameter choices and computation. They also provide open-source code and discuss implementation bottlenecks and possible acceleration strategies, underscoring the method’s robustness and potential for modeling diffusion-driven transport with complex boundary interactions.

Abstract

We formulate a finite-particle method of mass transport that accounts for general mixed boundary conditions. The particle method couples a geometrically-exact treatment of advection; Wasserstein gradient-flow dynamics; and a Kullback-Leibler representation of the entropy. General boundary conditions are enforced by introducing an adsorption/depletion layer at the boundary wherein particles are added or removed as dictated by the boundary conditions. We demonstrate the range and scope of the method through a number of examples of application, including absorption of particles into a sphere and flow through pipes of square and circular cross section, with and without occlusions. In all cases, the solution is observed to converge weakly, or in the sense of local averages.

Figures (16)

• Figure 1: Geometry of the spherical domain used for the verification of the method. The image shows the double boundary layer used to enforce the Dirichlet boundary conditions. The thickness of the boundary is assigned as $2b$, of which $b$ inside the sphere and $b$ outside the sphere.
• Figure 2: Sphere example. Time evolution of the relative total mass inside the sphere for various numbers of particles $n_\infty$.
• Figure 3: Sphere example. Time evolution of the relative polar inertia of the sphere for various numbers of particles $n_\infty$.
• Figure 4: Sphere example. Error in the $L^1$ norm of the total mass \ref{['eq-l1-sphere1']}. The broken line represents the best-fitting power law showing convergence (error $e_{M}\approx m_p^{0.38}$).
• Figure 5: Sphere example. Error in the $L^1$ norm of the total polar inertia \ref{['eq-l1-sphere2']}. The broken line represents the best-fitting power law showing convergence (error $e_{J_G}\approx m_p^{0.43}$).

Theorems & Definitions (2)

• Remark 4.1: $\rho$ is a measure
• Remark 4.2: Regularization