  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.
