A Minimally-Resolved Immersed Boundary Model for Reaction-Diffusion Problems

TL;DR

This paper introduces a minimally-resolved immersed boundary method for simulating reaction-diffusion processes in suspensions of reactive spherical particles, offering flexibility, efficiency, and accuracy at moderate densities without complex Green's function calculations.

Contribution

The authors develop a flexible, grid-based immersed boundary approach using reactive blobs that simplifies modeling of reaction-diffusion in particle dispersions, avoiding complex Green's functions.

Findings

Accurately models reaction-diffusion at low to moderate particle densities.

Computational cost comparable to solving a Poisson equation.

Validated against existing results for ordered and disordered suspensions.

Abstract

We develop an immersed-boundary approach to modeling reaction-diffusion processes in dispersions of reactive spherical particles, from the diffusion-limited to the reaction-limited setting. We represent each reactive particle with a minimally-resolved 'blob' using many fewer degrees of freedom per particle than standard discretization approaches. More complicated or more highly resolved particle shapes can be built out of a collection of reactive blobs. We demonstrate numerically that the blob model can provide an accurate representation at low to moderate packing densities of the reactive particles, at a cost not much larger than solving a Poisson equation in the same domain. Unlike multipole expansion methods, our method does not require analytically-computed Green's functions, but rather, computes regularized discrete Green's functions on the fly by using a standard grid-based discretization of the Poisson equation. This allows for great flexibility in implementing different boundary conditions, coupling to fluid flow or thermal transport, and the inclusion of other effects such as temporal evolution and even nonlinearities. We develop multigrid-based preconditioners for solving the linear systems that arise when using implicit temporal discretizations or studying steady states. In the diffusion-limited case the resulting linear system is a saddle-point problem, the efficient solution of which remains a challenge for suspensions of many particles. We validate our method by comparing to published results on reaction-diffusion in ordered and disordered suspensions of reactive spheres.