A decoupled and structure-preserving direct discontinuous Galerkin method for the Keller-Segel Model

In this work, we develop a novel numerical scheme to solve the classical Keller--Segel (KS) model which simultaneously preserves its intrinsic mathematical structure and achieves optimal accuracy. The model is reformulated into a gradient flow structure using the energy variational method, which reveals the inherent energy dissipative dynamics of the system. Based on this reformulation, we construct a structure-preserving discretization by semi-implicit method in time and the direct discontinuous Galerkin (DDG) method in space, resulting in a stable and high-order accurate approximation. The proposed scheme enjoys several desirable properties: (i) energy stability, ensuring discrete free energy dissipation; (ii) exact conservation of mass for the cell density; (iii) positivity preservation of the numerical cell density, enforced via a carefully designed limiter; and (iv) optimal convergence rate, with first-order accuracy in time and $(k+1)$-th order accuracy in space for polynomials of degree $k$. We provide rigorous theoretical analysis that substantiate these properties. In addition, extensive numerical experiments, including benchmark problems exhibiting pattern formation and near blow-up behavior, are conducted to validate the theoretical results and demonstrate the robustness, efficiency, and accuracy of the proposed method. The approach offers a flexible and reliable framework for structure-preserving numerical simulation of chemotaxis models and other gradient flow-type systems.