Communications on Applied Mathematics and Computation >

2024 , Vol. 6 >Issue 1: 372 - 398

DOI: https://doi.org/10.1007/s42967-023-00258-w

ORIGINAL PAPERS

High-Order Decoupled and Bound Preserving Local Discontinuous Galerkin Methods for a Class of Chemotaxis Models

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  • School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China

Received date: 2022-08-31

  Revised date: 2023-01-07

  Online published: 2024-04-16

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Research supported by the NSFC grant 12071455.

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Abstract

In this paper, we explore bound preserving and high-order accurate local discontinuous Galerkin (LDG) schemes to solve a class of chemotaxis models, including the classical Keller-Segel (KS) model and two other density-dependent problems. We use the convex splitting method, the variant energy quadratization method, and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models. These semi-implicit schemes are decoupled, energy stable, and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method. Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy. To overcome these difficulties, we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step. This bound preserving limiter results in the Karush-Kuhn-Tucker condition, which can be solved by an efficient active set semi-smooth Newton method. Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.

Key words： Chemotaxis models; Local discontinuous Galerkin (LDG) scheme; Convex splitting method; Variant energy quadratization method; Scalar auxiliary variable method; Spectral deferred correction method

Cite this article

Wei Zheng, Yan Xu . High-Order Decoupled and Bound Preserving Local Discontinuous Galerkin Methods for a Class of Chemotaxis Models[J]. Communications on Applied Mathematics and Computation, 2024 , 6(1) : 372 -398 . DOI: 10.1007/s42967-023-00258-w

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