Abstract
In this paper, we consider models of cancer migration and invasion, which consist of two nonlinear parabolic equations (one of the convection–diffusion reaction type and the other of the diffusion–reaction type) and an additional nonlinear ordinary differential equation. The unknowns represent concentrations or densities that cannot be negative. Widely used approximations, such as difference schemes, can produce negative solutions because of truncation errors and can become unstable. We propose a new difference scheme that guarantees the positivity of the numerical solution for arbitrary mesh step sizes. It has explicit and fast performance even for nonlinear reaction terms that consist of sums of positive and negative functions. The numerical examples illustrate the simplicity and efficiency of the method. A numerical simulation of a model of cancer migration is also discussed.
Highlights
Positivity-Preserving DifferenceTypically, a solid tumour starts to form when a mutated cell or cells are able to circumvent the normal regulatory functions of the body
The novelty of this paper is in the following: We develop an efficient nonstandard finite difference approximation, see, e.g., [35] for solving nonlinear systems of two parabolic and one ordinary differential equation
We investigate the consistency of the discretisations, and we shed light on the matter of convergence and preservation of the evolution and conservation laws of the differential problem by various numerical tests
Summary
Positivity-Preserving DifferenceTypically, a solid tumour starts to form when a mutated cell or cells are able to circumvent the normal regulatory functions of the body. Over the past few years, various mathematical models of tumour formation and growth have been proposed and analysed [2,3,4,5,6,7,8,9,10] (see the review paper [11]). The modelling of honey bee colonies has a direct relevance to developmental biology and studies of the effects of ecology and climate change, see, e.g., [12]. The model is formulated as an extension of Keller–Segel system with a sign-changing chemotactic coefficient. We consider the PDE systems modelling of tumour invasion through healthy tissue, which were introduced in [4] and developed in many papers, e.g., [5,6]
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