Numerical analysis of a semi-implicit Euler scheme for the Keller-Segel model

TL;DR

The paper analyzes a linear, decoupled semi-implicit Euler time discretization for the Keller-Segel model in 2D, applicable to both parabolic-elliptic and parabolic-parabolic forms. It proves that the scheme unconditionally preserves mass conservation, positivity of the cell density, and energy dissipation at the semi-discrete level, and establishes optimal $L^p$ error estimates for $1<p<\infty$. Under a small-mass condition and smoothness assumptions, it shows first-order convergence in time in $L^p$-norm for the numerical solutions $\rho^{n+1}$ and $c^{n+1}$, along with essential bounds on the chemoattractant component. The results provide rigorous theoretical guarantees for this widely used time discretization, forming a solid foundation for fully discrete, higher-order, or 3D extensions with preserved structural properties.

Abstract

We study the properties of a semi-implicit Euler scheme that is widely used in time discretization of Keller-Segel equations both in the parabolic-elliptic form and the parabolic-parabolic form. We prove that this linear, decoupled, first-order scheme preserves unconditionally the important properties of Keller-Segel equations at the semi-discrete level, including the mass conservation and positivity preserving of the cell density, and the energy dissipation. We also establish optimal error estimates in $L^p$-norm $(1<p<\infty)$.

Theorems & Definitions (21)

• Theorem 2.1
• proof
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Theorem 4.1