Convergence of mass transfer particle tracking schemes for the simulation of advection-diffusion-reaction equations

Abstract

Since their introduction [1], [2], multi-species mass-transfer particle tracking (or MTPT) algorithms have been used to accurately simulate advective and dispersive transport of solutes, even within systems that feature nonlinear chemical reactions. The MTPT methods were originally derived from a probabilistic or first-principles perspective and have previously lacked a more rigorous derivation arising directly from the underlying advection-diffusion-reaction equation (ADRE). Herein, we provide a fully rigorous derivation of the MTPT method as a Lagrangian approximation of solutions to the ADRE, complete with a description of the error and order of accuracy generated by this approximation. Numerical simulations further detail the fidelity of MTPT methods and display parameter regimes wherein the numerical error is independent of the chosen time step, and thereby dependent only upon the numerical discretization of the spatial domain via the number of particles within a simulation. Finally, different normalizations of the local Green's function are shown to generate similar approximations of the underlying solution for stationary particles.