  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 $10^{-3}$.
