Yekaterina Epshteyn

Department of Mathematical Sciences

Carnegie Mellon University

Pittsburgh, PA 15213

rina10@andrew.cmu.edu

Department of Mathematical Sciences

Carnegie Mellon University

Pittsburgh, PA 15213

rina10@andrew.cmu.edu

and

Alexander Kurganov

Mathematics Department

Tulane University

New Orleans, LA 70118

kurganov@math.tulane.edu

Mathematics Department

Tulane University

New Orleans, LA 70118

kurganov@math.tulane.edu

**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.

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