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 assume here the smallness of the mass of the initial data in order to ensure that the solutions of the original Keller-Segel equations do not blow up in finite time. We prove that this linear, decoupled, first-order scheme preserves 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 Lᵖ-norm (1<p<).
Goubet et al. (Sun,) studied this question.