An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and Invasion
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 conce...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/1/131 |
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| author | Mikhail K. Kolev Miglena N. Koleva Lubin G. Vulkov |
| author_facet | Mikhail K. Kolev Miglena N. Koleva Lubin G. Vulkov |
| author_sort | Mikhail K. Kolev |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-10T03:32:04Z |
| format | Article |
| id | doaj.art-a0ba9fe835404604ad70c7607d813f3a |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T03:32:04Z |
| publishDate | 2022-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-a0ba9fe835404604ad70c7607d813f3a2023-11-23T11:54:37ZengMDPI AGMathematics2227-73902022-01-0110113110.3390/math10010131An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and InvasionMikhail K. Kolev0Miglena N. Koleva1Lubin G. Vulkov2Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Słoneczna 54, 10-710 Olsztyn, PolandDepartment of Mathematics, University of Rousse, 8 Studentska St., 7017 Ruse, BulgariaDepartment of Mathematics, University of Rousse, 8 Studentska St., 7017 Ruse, BulgariaIn 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.https://www.mdpi.com/2227-7390/10/1/131cancer cells migrationfinite difference methodpositivity preservingevolutionconservation laws |
| spellingShingle | Mikhail K. Kolev Miglena N. Koleva Lubin G. Vulkov An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and Invasion Mathematics cancer cells migration finite difference method positivity preserving evolution conservation laws |
| title | An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and Invasion |
| title_full | An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and Invasion |
| title_fullStr | An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and Invasion |
| title_full_unstemmed | An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and Invasion |
| title_short | An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and Invasion |
| title_sort | unconditional positivity preserving difference scheme for models of cancer migration and invasion |
| topic | cancer cells migration finite difference method positivity preserving evolution conservation laws |
| url | https://www.mdpi.com/2227-7390/10/1/131 |
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