Establish rigorous convergence of the CRDME to the continuum model

Establish rigorous convergence, with quantitative mesh-width error estimates, of the density governed by the spatially discrete CRDME forward equation and of CRDME statistics to the corresponding continuum reaction-drift-diffusion model with two-body interaction potentials.

Background

The paper reports empirical second-order convergence as the mesh is refined, but the convergence of the discrete density to the continuum density—and even convergence of CRDME statistics to their continuum analogues—has not been established analytically for the full multiparticle model with interaction potentials.

The authors indicate that estimates from finite-element analyses of reaction-diffusion and reaction-drift-diffusion equations may provide ingredients for a proof. A rigorous result would validate the claimed convergence of the new CRDME beyond the numerical experiments and clarify the rate in the mesh width under appropriate regularity, mesh, potential, and reaction-kernel assumptions.