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Numerical solution of the two-phase tumour growth model with moving boundary

2019; Volume: 60; Linguagem: Inglês

10.21914/anziamj.v60i0.13936

1445-8810

Gopikrishnan Chirappurathu Remesan,

Mathematical and Theoretical Epidemiology and Ecology Models

A novel numerical technique is proposed to solve a two-phase tumour growth model in one spatial dimension without needing to account for the boundary dynamics explicitly. The equivalence to the standard definition of a weak solution is proved. The method is tested against equations with analytically known solutions, to illustrate the advantages over existing techniques. The tumour growth model is solved using the new procedure and is shown to be consistent with results available in the literature. References C. J. W. Breward, H. M. Byrne, and C. E. Lewis. The role of cell-cell interactions in a two-phase model for avascular tumour growth. J. Math. Biol., 45(2):125–152, 2002. doi:10.1007/s002850200149. C. J. W. Breward, H. M. Byrne, and C. E. Lewis. A multiphase model describing vascular tumour growth. B. Math. Biol., 65(4):609–640, 2003. doi:10.1016/S0092-8240(03)00027-2. H. M. Byrne, J. R. King, D. L. S. McElwain, and L. Preziosi. A two-phase model of solid tumour growth. Appl. Math. Lett., 16(4):567–573, 2003. doi:10.1016/S0893-9659(03)00038-7. L. C. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, 1998. URL https://www.ams.org/publications/authors/books/postpub/gsm-19-R. Eymard, Gallouet, and Herbin]eymard R. Eymard, T. Gallouet, and R. Herbin. Finite volume methods. In Solution of Equation in ℝn (Part 3), Techniques of Scientific Computing (Part 3), volume 7 of Handbook of Numerical Analysis, pages 713–1018. Elsevier, 2000. doi:10.1016/S1570-8659(00)07005-8. B. van Leer. Towards the ultimate conservative difference scheme. v. a second-order sequel to godunov's method. J. Comput. Phys., 32(1):101–136, 1979. doi:10.1016/0021-9991(79)90145-1. J. Ward and J. R. King. Mathematical modelling of avascular-tumour growth. Math. Med. Biol., 14(1):39–69, 1997. doi:10.1093/imammb/14.1.39. J. Ward and J. R. King. Mathematical modelling of avascular-tumor growth II: Modelling growth saturation. IMA J. Math. Appl. Med., 16:171–211, 1999. doi:10.1093/imammb16.2.171.