High-order positivity-preserving hybrid finite-volume-finite-difference methods for chemotaxis systems

Chemotaxis refers to mechanisms by which cellular motion occurs in response to an external stimulus, usually a chemical one. Chemotaxis phenomenon plays an important role in bacteria/cell aggregation and pattern formation mechanisms, as well as in tumor growth. A common property of all chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in rapid growth of solutions in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. There is consequently a need for accurate and computationally efficient numerical methods for the chemotaxis models. In this work, we develop and study novel high-order hybrid finite-volume-finite-difference schemes for the Patlak-Keller-Segel chemotaxis system and related models. We demonstrate high-accuracy, stability and computational efficiency of the proposed schemes in a number of numerical examples.

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