Finite Difference Approximation with ADI Scheme for Two-Dimensional Keller-Segel Equations (Communications in Computational Physics, 2024)
Keller-Segel systems are a set of nonlinear partial differential equations usedto model chemotaxis in biology. In this paper, we propose two alternating directionimplicit (ADI) schemes to solve the 2D Keller-Segel systems directly with minimalcomputational cost, while preserving positivity, energy dissipation law and mass conservation. One scheme unconditionally preserves positivity, while the other does soconditionally. Both schemes achieve second-order accuracy in space, with the formerbeing first-order accuracy in time and the latter second-order accuracy in time. Besides, the former scheme preserves the energy dissipation law asymptotically. We validate these results through numerical experiments, and also compare the efficiency ofour schemes with the standard five-point scheme, demonstrating that our approacheseffectively reduce computational costs.