An adaptive numerical scheme for a partial integro-differential equation

Abstract

One 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. 
 
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