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

Abstract

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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k+1)$$\end{document}-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.