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Keller-Segel systems are a set of nonlinear partial differential equationsused to model chemotaxis in biology. In this paper, we propose two alternatingdirection implicit (ADI) schemes to solve the 2D Keller-Segel systems directlywith minimal computational cost, while preserving positivity, energydissipation law and mass conservation. One scheme unconditionally preservespositivity, while the other does so conditionally. Both schemes achievesecond-order accuracy in space, with the former being first-order accuracy intime and the latter second-order accuracy in time. Besides, the former schemepreserves the energy dissipation law asymptotically. We validate these resultsthrough numerical experiments, and also compare the efficiency of our schemeswith the standard five-point scheme, demonstrating that our approacheseffectively reduce computational costs.

Finite Difference Approximation with ADI Scheme for Two-dimensional Keller-Segel Equations