High-accuracy positivity-preserving numerical method for Keller-Segel model

The Keller-Segel model is a time-dependent nonlinear partial differential system, which couples a reaction-diffusion-chemotaxis equation with a reaction-diffusion equation; the former describes cell density, and the latter depicts the concentration of chemoattractants. This model plays a vital role in the simulation of the biological processes. In view of the fact that most of the proposed numerical methods for solving the model are low-accuracy in the temporal direction, we aim to derive a high-precision and stable compact difference scheme by using a finite difference method to solve this model. First, a fourth-order backward difference formula and compact difference operators are respectively employed to discretize the temporal and spatial derivative terms in this model, and a compact difference scheme with the space-time fourth-order accuracy is proposed. To keep the accuracy of its boundary with the same order as the main scheme, a Taylor series expansion formula with the Peano remainder is used to discretize the boundary conditions. Then, based on the new scheme, a multigrid algorithm and a positivity-preserving algorithm which can guarantee the fourth-order accuracy are established. Finally, the accuracy and reliability of the proposed method are verified by diverse numerical experiments. Particularly, the finite-time blow-up, non-negativity, mass conservation and energy dissipation are numerically simulated and analyzed.