Commun. Comput. Phys., 10 (2011), pp. 474-508.


Local Discontinuous Galerkin Methods for the Degasperis-Procesi Equation

Yan Xu 1*, Chi-Wang Shu 2

1 Department of Mathematics, University of Science and Technology of China, Hefei 230026, Anhui, China.
2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.

Received 30 April 2010; Accepted (in revised version) 30 July 2010
Available online 28 April 2011
doi:10.4208/cicp.300410.300710a

Abstract

In this paper, we develop, analyze and test local discontinuous Galerkin (LDG) methods for solving the Degasperis-Procesi equation which contains nonlinear high order derivatives, and possibly discontinuous or sharp transition solutions. The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the L^2 stability for general solutions. The proof of the total variation stability of the schemes for the piecewise constant P^0 case is also given. The numerical simulation results for different types of solutions of the nonlinear Degasperis-Procesi equation are provided to illustrate the accuracy and capability of the LDG method.

AMS subject classifications: 65M60, 35Q53

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Key words: Local discontinuous Galerkin method, Degasperis-Procesi equation, L^2 stability, total variation stability.

*Corresponding author.
Email: yxu@ustc.edu.cn (Y. Xu), shu@dam.brown.edu (C.-W. Shu)
 

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