High-order explicit numerical methods for the one- and two-dimensional haptotaxis problems of tumor invasion

High-precision numerical methods are developed for biomathematical models that describe various biotaxis problems of tumor invasion, facilitating an in-depth exploration of the underlying evolutionary mechanism of tumor invasion. In this paper, we construct a high-order explicit numerical method for solving haptotaxis models of tumor invasion. To incorporate the specific characteristics of the initial and no-flux boundary conditions for the haptotaxis models, we employ a high-order compact finite difference method to discretize the spatial derivatives, thereby obtaining a series of semi-discrete ordinary differential systems to approximate the solutions of these models. The strong stability-preserving (SSP) Runge-Kutta method is utilized in the semi-discrete ordinary differential systems to obtain the third-order accuracy in time. A high-order numerical integration method is used to design a positivity-preserving algorithm with spatially fourth-order accuracy guaranteed. The designed explicit method not only has high accuracy and good stability, but also has obvious positivity-preserving effect, which is more accurate and reliable than the numerical methods for this problem in the existing literature. To validate the performances of the proposed methods, several numerical examples are provided, and all sorts of complicated biotaxis dynamics for the haptotaxis models are simulated.