Numerical solution of a malignant invasion model using some finite difference methods

In this article, one standard and four nonstandard finite difference methods are used to solve a cross-diffusion malignant invasion model. The model consists of a system of nonlinear coupled partial differential equations (PDEs) subject to specified initial and boundary conditions, and no exact solution is known for this problem. It is difficult to obtain theoretically the stability region of the classical finite difference scheme to solve the set of nonlinear coupled PDEs, this is one of the challenges of this class of method in this work. Three nonstandard methods abbreviated as NSFD1, NSFD2, and NSFD3 are considered from the study of Chapwanya et al., and these methods have been constructed by the use of a more general function replacing the denominator of the discrete derivative and nonlocal approximations of nonlocal terms. It is shown that NSFD1, which preserves positivity when used to solve classical reaction-diffusion equations, does not inherit this property when used for the cross-diffusion system of PDEs. NSFD2 and NSFD3 are obtained by appropriate modifications of NSFD1. NSFD2 is positivity-preserving when the functional relationship [ψ(h)]2=2ϕ(k){\left[\psi \left(h)]}^{2}=2\phi \left(k) holds, while NSFD3 is unconditionally dynamically consistent with respect to positivity. First, we show that NSFD2 and NSFD3 are not consistent methods. Second, we tried to modify NSFD2 in order to make it consistent but we were not successful. Third, we extend NSFD3 so that it becomes consistent and still preserves positivity. We denote the extended version of NSFD3 as NSFD5. Finally, we compute the numerical rate of convergence in time for NSFD5 and show that it is close to the theoretical value. NSFD5 is consistent under certain conditions on the step sizes and is unconditionally positivity-preserving.