Notes on finite difference schemes to a parabolic-elliptic system modelling chemotaxis

We consider numerical solutions to a nonlinear parabolic-elliptic system in mathematical biology. An important property of solutions to the system is the conservation of L1 norm. Applying the upwind technique, we make a finite difference scheme satisfying the conservation of a discrete L1 norm. We also describe a remark on time discretization from the view point of Lyapunov's property which is another important feature of the system. Several numerical examples clarify the nature of our scheme.