Discrete-to-continuum limit for nonlinear reaction-diffusion systems via EDP convergence for gradient systems
Georg Heinze, Alexander Mielke, Artur Stephan
TL;DR
The paper develops a rigorous discrete-to-continuum transition for nonlinear reaction-diffusion systems with mass-action and detailed balance by casting both the discrete and continuum dynamics as gradient systems in a continuity-equation framework. It introduces two parallel gradient structures, one on a spatial lattice and one on the continuous torus, linked through carefully constructed embeddings and an energy-dissipation principle (EDP) to ensure convergence. Central to the analysis are a cosh-based dissipation structure, a magical estimate for the perspective function, and a robust compactness/liminf framework that handles non-convex dissipation and nonlinear reactions; the authors prove discrete chain rules and a continuum chain rule, and establish convergence of embedded discrete solutions to continuum EDB solutions under growth constraints. This variational, metric-gradient approach provides a rigorous foundation for discrete approximations of nonlinear RD systems and extends gradient-flow convergence theory beyond linear cases, with implications for numerical analysis and model reduction in reaction-diffusion contexts.
Abstract
We investigate the convergence of spatial discretizations for reaction-diffusion systems with mass-action law satisfying a detailed balance condition. Considering systems on the d-dimensional torus, we construct appropriate space-discrete processes and show convergence not only on the level of solutions, but also on the level of the gradient systems governing the evolutions. As an important step, we prove chain rule inequalities for the reaction-diffusion systems as well as their discretizations, featuring a non-convex dissipation functional. The convergence is then obtained with variational methods by building on the recently introduced notion of gradient systems in continuity equation format.