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.

在本文中，我们采用带界面校正的直接不连续Galerkin方法（DDGIC）求解趋化性Keller-Segel方程，并证明满足正定性的二次多项式解具有三阶精度。我们显示DDGIC方法可以为细胞密度变量获得最佳收敛阶，而无需引入额外的辅助变量来近似化学浓度梯度。这是由于DDGIC方法的超收敛性。我们证明了，用数字流量系数的正确选择，细胞密度近似可以在任何时间保存无负。使用保持阳性的限制器可保持一致的三阶精度。对于具有爆炸或奇异解的趋化模型，