Validation and Calibration of Models for Reaction-Diffusion Systems

TL;DR

This paper introduces a new class of explicit difference methods for reaction-diffusion models that naturally incorporate space and time scales, minimize anisotropy effects, and provide highly accurate numerical solutions for pattern formation.

Contribution

The authors develop a novel explicit difference scheme with fixed parameter dependence on space and time scales, reducing anisotropy effects and enabling precise calibration of diffusion models.

Findings

Numerical solutions approach the exact solution as $( riangle x)^{2(N+2)}$

Discretization errors are around $10^{-6}$ in the sup norm

Circular wave patterns deviate less than 0.2% from perfect symmetry

Abstract

Space and time scales are not independent in diffusion. In fact, numerical simulations show that different patterns are obtained when space and time steps ($\Delta x$ and $\Delta t$) are varied independently. On the other hand, anisotropy effects due to the symmetries of the discretization lattice prevent the quantitative calibration of models. We introduce a new class of explicit difference methods for numerical integration of diffusion and reaction-diffusion equations, where the dependence on space and time scales occurs naturally. Numerical solutions approach the exact solution of the continuous diffusion equation for finite $\Delta x$ and $\Delta t$, if the parameter $\gamma_N=D \Delta t/(\Delta x)^2$ assumes a fixed constant value, where $N$ is an odd positive integer parametrizing the alghorithm. The error between the solutions of the discrete and the continuous equations goes to zero as $(\Delta x)^{2(N+2)}$ and the values of $\gamma_N$ are dimension independent. With these new integration methods, anisotropy effects resulting from the finite differences are minimized, defining a standard for validation and calibration of numerical solutions of diffusion and reaction-diffusion equations. Comparison between numerical and analytical solutions of reaction-diffusion equations give global discretization errors of the order of $10^{-6}$ in the sup norm. Circular patterns of travelling waves have a maximum relative random deviation from the spherical symmetry of the order of 0.2%, and the standard deviation of the fluctuations around the mean circular wave front is of the order of $10^{-3}$.