Keller-Segel方程式の数値解析(<特集>移流項をもつ反応拡散系)

抄録

We are concerned with numerical methods for the Keller-Segel system that describes the aggregation of slime molds resulting from their chemotactic features. Whereas the system has positivity and mass conservation properties, it is not certain that numerical schemes for the system preserve these conservation properties. In the present article, we review two conservative numerical schemes proposed by the author and discuss how to choose the time increments and the space meshes in order to realize those conservation properties. The first one is the finite-difference method that makes use of the semi-implicit time discretization with the time-increment control and the upwind difference approximation. The second is the finite-element method that is an application of Baba-Tabata's conservative upwind finite element. Conservative properties are proved via the theory of M-matrices. Error analysis of the finite-element scheme is also summarized. In particular, we have error estimates with explicit convergence rates by virtue of the analytical semigroup theory.