Validation and Calibration of Models for Reaction-Diffusion Systems

Space and time scales are not independent in diffusion. In fact, numerical simulations show that different patterns are obtained when space and time steps (∆x and ∆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 ∆x and ∆t, if the parameter γN = D∆t/(∆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 (∆x) and the values of γN 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 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. 1 [email protected] 2 [email protected] Phone: (351)-(1)-8417617, FAX: (351)-(1)-8419123