Numerical interactions of random and directed motility during cancer invasion

The continuum modelling of cell migration during cancer invasion results in the coupling of parabolic and hyperbolic partial differential equations (PDEs) arising from the random motility of normal tissue and the directed movement up substrate gradients of malignant cells. The numerical solution of such systems of equations require different stability criteria being simultaneously satisfied. We show that in such a coupled system, the origins of numerical instability can be identified by analyzing the fastest growing mode in a numerically unstable solution. In general, stability can be achieved by choosing an appropriate grid size representing the more stringent of the conditions for hyperbolic and parabolic stability. However, this induces variable degrees of numerical diffusion because of a changing CFL (Courant, Friedrichs, and Lewy) number. Solving the hyperbolic and parabolic PDEs on separate grids results in a better convergence of the solution. Finally, we discuss the use of higher-order schemes for the solution of such problems. Cancer modelling brings together directed and random motility in a unique way thereby presenting interesting new numerical problems.