Abstract:
We design nonstandard finite difference (NSFD) schemes which are unconditionally dynamically
consistent with respect to the positivity property of solutions of cross-diffusion
equations in biosciences. This settles a problem that was open for quite some time. The
study is done in the setting of three concrete and highly relevant cross-diffusion systems
regarding tumor growth, convective predator–prey pursuit and evasion model and
reaction–diffusion–chemotaxis model. It is shown that NSFD schemes used for classical
reaction–diffusion equations, such as the Fisher equation, for which the solutions enjoy
the positivity property, are not appropriate for cross-diffusion systems. The reliable NSFD
schemes are therefore obtained by considering a suitable implementation on the crossdiffusive
term of Mickens’ rule of nonlocal approximation of nonlinear terms, apart from
his rule of complex denominator function of discrete derivatives. We provide numerical
experiments that support the theory as well as the power of the NSFD schemes over the
standard ones. In the case of the cancer growth model, we demonstrate computationally
that our NSFD schemes replicate the property of traveling wave solutions of developing
shocks observed in Marchant et al. (2000).
