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 ($\Delta x$ 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 $10^{-3}$.Comment: 33 pages, 8 figures, to appear in Int. J. Bifurcation and Chao

Lattice Gas Automata for Reactive Systems

Reactive lattice gas automata provide a microscopic approachto the dynamics of spatially-distributed reacting systems. After introducing the subject within the wider framework of lattice gas automata (LGA) as a microscopic approach to the phenomenology of macroscopic systems, we describe the reactive LGA in terms of a simple physical picture to show how an automaton can be constructed to capture the essentials of a reactive molecular dynamics scheme. The statistical mechanical theory of the automaton is then developed for diffusive transport and for reactive processes, and a general algorithm is presented for reactive LGA. The method is illustrated by considering applications to bistable and excitable media, oscillatory behavior in reactive systems, chemical chaos and pattern formation triggered by Turing bifurcations. The reactive lattice gas scheme is contrasted with related cellular automaton methods and the paper concludes with a discussion of future perspectives.Comment: to appear in PHYSICS REPORTS, 81 revtex pages; uuencoded gziped postscript file; figures available from [email protected] or [email protected]