A decoupled and structure-preserving direct discontinuous Galerkin method for the Keller-Segel Model
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| Format: | Preprint |
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2025
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| author | Yin, X. Lan, X. Qin, Y. |
| author_facet | Yin, X. Lan, X. Qin, Y. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_16940 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A decoupled and structure-preserving direct discontinuous Galerkin method for the Keller-Segel Model Yin, X. Lan, X. Qin, Y. Numerical Analysis 65M60 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. |
| title | A decoupled and structure-preserving direct discontinuous Galerkin method for the Keller-Segel Model |
| topic | Numerical Analysis 65M60 |
| url | https://arxiv.org/abs/2509.16940 |