  In this article we propose a unified framework in order to study
reaction-diffusion systems containing self- and cross-diffusion using a free
energy approach. This framework naturally leads to the formulation of an energy
law, and to a numerical method respecting a discrete version of the latter. It
constitutes an alternative method and complements the standard linear stability
analysis, as it allows for the numerical study of nonlinear patterns, while
monitoring the energy evolution, even in complex geometries. As an application,
we propose and study a modified Gray-Scott system augmented with self- and
cross-diffusion terms. Numerical simulations unveil original patterns, clearly
distinct from those obtained with linear diffusion only
