Local Discontinuous Galerkin Methods for the Degasperis-Procesi Equation
Yan Xu 1*, Chi-Wang Shu 21 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
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.
Email: firstname.lastname@example.org (Y. Xu), email@example.com (C.-W. Shu)