High-order mass conserving, positivity plus energy-law preserving schemes and their error estimates for Keller-Segel equations

TL;DR

This paper introduces high-order, mass-conserving, positivity-preserving, and energy-law preserving numerical schemes for the Keller-Segel equations, with rigorous error analysis and numerical validation.

Contribution

It develops novel high-order linear and decoupled schemes that simultaneously preserve key physical properties of the Keller-Segel model, including mass, positivity, and energy law.

Findings

Schemes successfully preserve mass, positivity, and energy law.

Optimal error estimates are rigorously derived.

Numerical experiments confirm the schemes' effectiveness.

Abstract

Chemotaxis plays a significant role in numerous physiological processes. The Keller-Segel equation serves as a mathematical model for simulating the phenomenon of cell population aggregation under chemotaxis, possessing physical properties such as mass conservation, positivity of density, and energy dissipation. High-order linear and decoupled schemes for the parabolic-parabolic Keller-Segel chemotaxis model are proposed in this paper, which satisfy the three physical properties mentioned earlier. Firstly, by applying a logarithmic transformation, the Keller-Segel model is reformulated into its equivalent form that maintains the positivity of cell density regardless of the discrete scheme. Based on this equivalent system, we then propose high-order linear and decoupled numerical schemes using the backward differentiation formula (BDF). Furthermore, through the incorporation of a recovery technique and an energy-law preservation correction (EPC), we ensure that these schemes maintain mass conservation and preserve the original energy-law. Finally, we conduct a rigorous optimal error analysis for the numerical schemes under certain assumptions regarding the regularity of solutions, and some numerical experiments are also presented to demonstrate their effectiveness.