# The local discontinuous Galerkin method with generalized alternating flux for solving Burger's equation

Volume 12, Issue 5, pp 300--313 Publication Date: December 20, 2018       Article History
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### Authors

Rongpei Zhang - School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang 110034, P. R. China. Di Wang - School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang 110034, P. R. China. Xijun Yu - Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, P. R. China. Bo Chen - College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, P. R. China. Zhen Wang - College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, P. R. China.

### Abstract

In this paper, we propose the local discontinuous Galerkin method based on the generalized alternating numerical flux for solving the one-dimensional nonlinear Burger's equation with Dirichlet boundary conditions. Based on the Hopf-Cole transformation, the original equation is transformed into a linear heat conduction equation with homogeneous Neumann boundary conditions. We will show that this method preserves stability. By virtue of the generalized Gauss-Radau projection, we can obtain the sub-optimal rate of convergence in $L^2$-norm of $\mathcal{O}(h^{k+\frac{1}{2}})$ with polynomial of degree $k$ and grid size $h$. Numerical experiments are given to verify the theoretical results.

### Keywords

• Burger's equation
• local discontinuous Galerkin method
• Hopf-Cole transformation
• generalized alternating numerical flux

•  65M60

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