New Interior Penalty Discontinuous Galerkin Methods for the Keller–Segel Chemotaxis Model

Abstract

We develop a family of new interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model. This model is described by a system of two nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. It has been recently shown that the convective part of this system is of a mixed hyperbolic-elliptic type, which may cause severe instabilities when the studied system is solved by straightforward numerical methods. Therefore, the first step in the derivation of our new methods is made by introducing the new variable for the gradient of the chemoattractant concentration and by reformulating the original Keller-Segel model in the form of a convection-diffusion-reaction system with a hyperbolic convective part. We then design interior penalty discontinuous Galerkin methods for the rewritten Keller-Segel system. Our methods employ the central-upwind numerical fluxes, originally developed in the context of finite-volume methods for hyperbolic systems of conservation laws.In this paper, we consider Cartesian grids and prove error estimates for the proposed high-order discontinuous Galerkin methods. Our proof is valid for pre-blow-up times since we assume boundedness of the exact solution. We also show that the blow-up time of the exact solution is bounded from above by the blow-up time of our numerical solution. In the numerical tests presented below, we demonstrate that the obtained numerical solutions have no negative values and are oscillation-free, even though no slope limiting technique has been implemented.

Fully Discrete Analysis of a Discontinuous Finite Element Method for the Keller-Segel Chemotaxis Model

“…The underlying spatial discretization of the model is based on the methods proposed recently in [18] and the temporal discretization is based either on Forward Euler or the second order explicit total variation diminishing (TVD) Runge-Kutta methods.…”

“…However, the operator splitting approach may not be applicable when a convective part of the chemotaxis system is not hyperbolic, which is a generic situation for the original Keller-Segel model as it was shown in [10], where the finitevolume Godunov-type central-upwind scheme was derived for (1.1) and extended to some other chemotaxis and haptotaxis models. The high-order discontinuous Galerkin method that is investigated here is based on the method proposed in [18]. The DG methods have recently become increasingly popular thanks to their flexibility for adaptive simulations, suitability for parallel computations, applicability to problems with discontinuous coefficients and/or solutions, and compatibility with other numerical methods.…”

“…In order to develop high-order DG methods for (1.1) in [18], the Keller-Segel system is rewritten as a system of the nonlinear convection-diffusion-reaction equations by introducing new variables (u, v) := ∇c :…”

“…The methods proposed in [18] are based on three primal DG methods: the Nonsymmetric Interior Penalty Galerkin (NIPG), the Symmetric Interior Penalty Galerkin (SIPG), and the Incomplete Interior Penalty Galerkin (IIPG) methods, [3,16,36]. The numerical fluxes in the proposed DG methods are the fluxes developed for the semidiscrete finite-volume central-upwind schemes in [30] (see also [29,31]).…”

“…At the same time, a certain upwinding information-one-sided speeds of propagation-is incorporated into the central-upwind fluxes. In [18], Cartesian grids are considered and the continuous in time error estimates are proved for the proposed high-order DG methods under the assumption of boundedness of the exact solution. Some numerical tests that validate the methods are considered in [18] and [17].…”