An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and Invasion

Mikhail K. Kolev, Miglena N. Koleva and Lubin G. Vulkov
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Mikhail K. Kolev: Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Słoneczna 54, 10-710 Olsztyn, Poland
Miglena N. Koleva: Department of Mathematics, University of Rousse, 8 Studentska St., 7017 Ruse, Bulgaria
Lubin G. Vulkov: Department of Mathematics, University of Rousse, 8 Studentska St., 7017 Ruse, Bulgaria

Mathematics, 2022, vol. 10, issue 1, 1-22

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.

Keywords: cancer cells migration; finite difference method; positivity preserving; evolution; conservation laws (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (1)

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