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

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