An adaptive numerical scheme for a partial integro-differential equation
Keywords:partial integro-differential, equation, numerical, adaptive,
AbstractOne method of modelling cell-cell adhesion gives rise to a partial integro-differential equation. While non-adaptive techniques work in the numerical modelling of such an equation, there are also many opportunities for optimisation. The studied partial integro-differential equation has a tendency to produce aggregations leaving large regions where both the function value and derivative are equal to zero, leading to a higher resolution than needed and lower than desired resolution where the aggregations form. In order to overcome this we develop an adaptive scheme in both space and time using a modified form of Matlab's ode45 and finite volume methods to more efficiently simulate the studied partial integro-differential equation. We use our numerical scheme to simulate the problem presented by Armstrong et al. [J. Theor. Biol. 243 (2006), pp. 98--113] and compare results. References
- N. J. Armstrong, K. J. Painter, and J. A. Sherratt. A continuum approach to modelling cell-cell adhesion. J. Theor. Biol., 243:98–113, 2006. doi:10.1016/j.jtbi.2006.05.030.
- R. J. LeVeque. Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, 2002. doi:10.1017/CBO9780511791253.
- C. F. Van Loan. Introduction to scientific computing: A matrix vector approach using MATLAB. MATLAB Curriculum. Prentise Hall, 1997.
- J. A. Sheratt, S. A. Gourley, N. J. Armstrong, and K. J. Painter. Boundedness of solutions of a non-local reaction-diffusion model for adhesion in cell aggregation and cancer invasion. Eur. J. Appl. Math., 20(1):123–144, 2009. doi:10.1017/S0956792508007742.
- M. S. Steinberg. On the mechanism of tissue reconstruction by dissociated cells, I. Population kinetics, differential adhesiveness and the absence of directed migration. P. Natl Acad. Sci. USA, 48(9):1577–1582, 1962a. doi:10.1073/pnas.48.9.1577.
- M. S. Steinberg. Mechanism of tissue reconstruction by dissociated cells, II: Time-course of events. Science, 137(3532):762–763, 1962b. doi:10.1126/science.137.3532.762.
- M. S. Steinberg. On the mechanism of tissue reconstruction by dissociated cells, III. Free energy relations and the reorganisation of fused, heteronomic tissue fragments. P. Natl Acad. Sci. USA, 48(10):1769–1776, 1962c. doi:10.1073/pnas.48.10.1769.
- P. L. Townes and J. Holtfreter. Directed movements and selective adhesion of embryonic amphibian cells. J. Exp. Zool., 128(1):53–120, 1955. doi:10.1002/jez.1401280105.
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