A fast and unconditionally positive finite-difference discretization of a multidimensional equation in nonlinear population dynamics

Abstract

Departing from a nonlinear diffusion–reaction equation which describes the growth of biological films, we derive a finite-difference discretization which preserves unconditionally the positivity and the boundedness of approximations. The design of this method follows a non-traditional approach in the estimation of first-order partial derivatives, and the technique is a variable step-size and exact methodology for which the properties of existence and uniqueness of non-negative and bounded solutions hold for any initial profile which is likewise non-negative and bounded. As a consequence of the exactness, the computer implementation requires less resources and yields faster results than some linear schemes available in the literature. Qualitative comparisons against known linear and nonlinear techniques show that our method produces similar computer results in the two-dimensional case. Moreover, our scheme is also able to simulate the development of microbial films in the three-dimensional scenario. This is a feature that is not inherent to the linear methodologies considered, in view of the large amount of computational resources that such approaches require.