A Positivity-Preserving Fourier Spectral Moving Mesh Method for the Keller-Segel Chemotaxis Model

Abstract

We develop a numerical  method for the Keller-Segel chemotaxis system that is designed to
(i) preserve the model’s fundamental structural properties (positivity / bound preservation, mass conservation, and energy dissipation),
(ii) efficiently and accurately resolve the near-singular dynamics associated with spike formation and finite-time blow-up.
Our approach combines a linear, positivity-preserving scalar auxiliary variable (SAV) scheme (following the framework in [15]) with a Fourier spectral spatial discretization and an moving-mesh PDE-based method. The SAV reformulation provides a convenient platform for stable, linear time stepping while maintaining energy dissipation; the Fourier spectral discretization delivers high accuracy in smooth regions; and the moving-mesh PDE mesh redistribution concentrates collocation points in regions of large gradients so that sharp, localized structures can be resolved without prohibitive cost. We show that the proposed moving mesh SAV scheme inherits positivity preservation, mass conservation, and discrete energy dissipation provided the mesh motion avoids element overlap. Two-dimensional tests demonstrate the method’s ability to capture fine spike profiles and estimate blow-up times with substantially reduced computational effort; the formulation extends straightforwardly to three spatial dimensions. Numerical results show that the proposed method is a practical and effective method for accurate simulation of chemotactic aggregation.