Chemical reaction-diffusion networks; convergence of the method of lines

TL;DR

This paper proves that solutions of a reaction-diffusion PDE system can be approximated by a discretized ODE system modeling chemical reactions in subdomains, with convergence shown via a consistency estimate.

Contribution

It introduces a space discretization method for reaction-diffusion systems and proves its convergence, applicable to various reaction networks and multiple spatial dimensions.

Findings

Convergence of the discretized ODE system to the PDE solution in L^2 norm.

The method is generalizable to other reaction networks and higher dimensions.

The approach connects reaction-diffusion PDEs with compartmental ODE models.

Abstract

We show that solutions of the chemical reaction-diffusion system associated to $A+B\rightleftharpoons C$ in one spatial dimension can be approximated in $L^2$ on any finite time interval by solutions of a space discretized ODE system which models the corresponding chemical reaction system replicated in the discretization subdomains where the concentrations are assumed spatially constant. Same-species reactions through the virtual boundaries of adjacent subdomains lead to diffusion in the vanishing limit. We show convergence of our numerical scheme by way of a consistency estimate, with features generalizable to reaction networks other than the one considered here, and to multiple space dimensions. In particular, the connection with the class of complex-balanced systems is briefly discussed here, and will be considered in future work.