Effects of different discretisations of the Laplacian upon stochastic simulations of reaction-diffusion systems on both static and growing domains (Q2029642)

The purpose of this paper is an investigation of the effects of different methods of the derivation of diffusive jump rates and different domain discretizations on stochastic reaction-diffusion systems modeled using the reaction-diffusion master equation framework. The authors extended and applied the results in [\textit{L. Meinecke} and \textit{P. Lötsted} et al., ibid. 294, 1--11 (2016; Zbl 1327.65019)] to a number of reaction-diffusion systems, including a range of reaction types, on both static and growing domains. These derivations are based on three different numerical schemes: the finite difference method (FDM), the finite volume method (FVM), the finite element method (FEM). The FVM generally gives for the production-decay examples, consisting solely of zeroth and first-order reactions, the most reliable results in terms of the error between stochastic simulations and solution of the corresponding macroscale PDE. Furthermore, the effects of different methods of the derivation of diffusive jump rates on pattern formation are studied, using the Turing reaction-diffusion model with both Schnakenberg and Gray-Scott kinetics as examples. The authors also investigated the effects of uniform domain growth on derivations of the diffusive jump rates, and simulations of pattern formation for the three discretization methods FDM, FVM, and FEM. It was shown for the Gray-Scott system that the choice of method of derivation of the jump rates can significantly impact the formation of patterns.