Finite volume methods for a Keller-Segel system: discrete energy, error estimates and numerical blow-up analysis
The authors applies the finite volume method to approximately solve the parabolic-elliptic system of the form \[ \frac{\partial u}{\partial t}=\nabla(\nabla u- u\nabla v), \] \[ -\Delta v + v=u \text{ in }\Omega \times [0,T], \,\Omega \subset \mathbb R^{2}, \] with the initial condition \( u(x,0)=u_{0}(x)\) on \(\Omega \) and homogenous boundary conditions for the fist derivatives of \(u\) and\(v\) on \( \Gamma =\partial \Omega.\) To preserve the essential properties of the exact problem -- the conservation laws and free energy -- several conditions are imposed in the construction of the mesh and adequate approximate scheme. The error estimate and blow-up phenomenon for approximate solutions are shown by numerical results for two examples on the rectangular and circular domains.
scientific article; zbMATH DE number 6677840
scientific article