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

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 $(1<p<\infty)$.