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

Volume 12, Issue 5, pp 300--313 http://dx.doi.org/10.22436/jnsa.012.05.04

Publication Date: December 20, 2018 Submission Date: January 15, 2018 Revision Date: August 04, 2018 Accteptance Date: October 06, 2018

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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.

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Keywords

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

MSC

•  65M60

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