Convergence of a cellular automata reaction-diffusion model to the PDE model

Cellular automata (CA) are used to simulate physical processes with various degrees of precision, but the theoretical quantitative bounds for this precision are rarely computed. We create a stochastic CA model of reaction-diffusion process with a parameter that can increase its precision by increasing the number of molecules within. We convert the solution of this CA to a piecewise-constant function and compare it with a regular PDE solution with similar initial conditions. The main result of presented work is that, as the precision parameter increases, the CA solution converges in mean square to a certain deterministic numerical scheme, which converges to the PDE solution in the limit. This convergence may also be achieved for different stochastic CA under some conditions.