Semi-implicit finite volume schemes for a chemotaxis-growth model

This paper is concerned with finite volume approximations for a nonlinear parabolic–elliptic system for chemotaxis-growth in RdRd, d=2,3d=2,3. This model describes a process of pattern formation by some chemotactic biological individuals. We present two schemes which make use of a semi-implicit time discretization and an upwind finite volume approximation. For both schemes, we prove existence, uniqueness and nonnegativity of the approximate solutions under some conditions on the time step, and we show (for one of the schemes) that the numerical solution converges to a corresponding weak solution for the studied model. Numerical simulations are performed in two dimensional spaces to demonstrate the efficiency of the schemes to capture the pattern formations and to verify our theoretical results.