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

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.