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 L-1 norm. Applying the upwind technique, we make a finite difference scheme satisfying the conservation of a discrete L-1 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. (c) 2005 Elsevier Inc. All rights reserved.