Positivity-preserving high-order compact difference method for the Keller-Segel chemotaxis model.

Area of Science:

• Mathematical Biology
• Computational Mathematics
• Numerical Analysis

Background:

• The Keller-Segel model describes chemotaxis, a fundamental biological process.
• Accurate and stable numerical methods are crucial for simulating complex biological phenomena.
• Existing methods may struggle with preserving non-negativity of cell density.

Purpose of the Study:

• To develop a high-order accurate, positivity-preserving numerical method for the Keller-Segel chemotaxis model.
• To ensure the non-negativity of cell density throughout simulations without compromising accuracy.
• To provide a robust computational tool for studying chemotaxis.

Main Methods:

• Development of a stiffly-stable, five-step, fourth-order fully implicit compact difference scheme.
• Implementation of a computational strategy for the nonlinear chemotaxis term.
• Design of a positivity-preserving numerical algorithm and a time advancement algorithm.

Main Results:

• The proposed scheme achieves fourth-order accuracy in both spatial and temporal directions.
• The numerical algorithm guarantees non-negativity of cell density at all time steps.
• Numerical simulations demonstrate the method's accuracy, stability, and positivity-preserving properties.

Conclusions:

• The developed compact difference method is accurate, stable, and effectively preserves positivity.
• This method provides a reliable approach for numerical simulations of chemotaxis phenomena.
• The findings contribute to advancing computational methods in mathematical biology.