  Chemically-active colloids modify the concentration of chemical solutes
surrounding them in order to self-propel. In doing so, they generate
long-ranged hydrodynamic flows and chemical gradients that modify the
trajectories of other particles. As a result, the dynamics of reactive
suspensions is fundamentally governed by hydro-chemical interactions. A full
solution of the detailed hydro-chemical problem with many particles is
challenging and computationally expensive. Most current methods rely on the
Green's functions of the Laplace and Stokes operators to approximate the
particle signatures in the far-field, which is only valid in the very dilute
limit in simple geometries. To overcome these limitations, we propose a
regularized mutipole framework, directly inspired by the Force Coupling Method
(FCM), to model phoretic suspensions. Our approach, called Diffusio-phoretic
FCM (DFCM), relies on grid-based volume averages of the concentration field to
compute the particle surface concentration moments. These moments define the
chemical multipoles of the diffusion (Laplace) problem and provide the swimming
forcing of the Stokes equations. Unlike far-field models based on singularity
superposition, DFCM accounts for mutually-induced dipoles. The accuracy of the
method is evaluated against exact and accurate numerical solutions for a few
canonical cases. We also quantify its improvements over far-field
approximations for a wide range of inter-particle distances. The resulting
framework can readily be implemented into efficient CFD solvers, allowing for
large scale simulations of semi-dilute diffusio-phoretic suspensions.
