Chemical reaction-diffusion networks; convergence of the method of lines
Provides a rigorous convergence proof for a numerical scheme linking reaction-diffusion PDEs to ODEs, relevant for computational chemistry and mathematical biology.
The paper proves that solutions of a chemical reaction-diffusion system for A+B⇌C in 1D can be approximated in L² by a space-discretized ODE system, with convergence shown via a consistency estimate. The method generalizes to other reaction networks and multiple dimensions.
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.