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

We study the properties of a semi-implicit Euler scheme widely used for time discretization of Keller-Segel equations in both parabolic-elliptic and parabolic-parabolic forms. We assume the initial mass of the cell density is sufficiently small to ensure that solutions of the continuous Keller-Segel equations exist globally in time. We prove that this linear, decoupled, first-order scheme preserves key properties of the Keller-Segel model at the semi-discrete level, including mass conservation, positivity preservation, and energy dissipation. We also establish optimal error estimates in Lp-norm (1 < p < ∞) for the scheme.