Finite volume methods for a Keller–Segel system: discrete energy, error estimates and numerical blow-up analysis

Abstract

We consider the finite volume approximation for a non-linear parabolic-elliptic system, which describes the aggregation of slime molds resulting from their chemotactic features, called a simplified Keller---Segel system. First, we present a linear finite volume scheme that satisfies both positivity and mass conservations, which are important features of the original system. We derive some inequalities on the discrete free energy. Then, under some assumptions on the regularity of solution, admissible mesh and a priori estimates of the discrete solution, we establish error estimates in $$L^p$$Lp norm with a suitable $$p>2$$p>2 for the two dimensional case. In the last part of this paper, we restrict our attention to the radially symmetric solution of chemotaxis system, and we derive some inequalities concerned with the blow-up phenomenon of numerical solution. Several numerical experiments are presented to verify the theoretical results.