Third order positivity-preserving direct discontinuous Galerkin method with interface correction for chemotaxis Keller-Segel equations

Abstract

In this article, we apply direct discontinuous Galerkin method with interface correction (DDGIC) to solve chemotaxis Keller-Segel equation and prove the quadratic polynomial solution satisfying positivity-preserving with third order of accuracy. We show DDGIC method can obtain optimal convergence order for the cell density variable without introducing extra auxiliary variables to approximate the gradient of the chemical concentration. This is due to the super convergent property of the DDGIC method. We prove that, with the proper choice of the numerical flux coefficients, the cell density approximation can be preserved none negative at all time. Uniform third order of accuracy is maintained with the positivity-preserving limiter applied. For the chemotaxis model with blow up or singular solutions, the DDGIC method can effectively remove negative values approximations and accurately capture the blow up time.