Methods and Applications of Analysis

Volume 21 (2014)

Number 1

A conservative discontinuous Galerkin method for the Degasperis-Procesi equation

Pages: 67 – 90

DOI: https://dx.doi.org/10.4310/MAA.2014.v21.n1.a3

Authors

Yunqing Huang (Hunan Key Laboratory for Computation and Simulation in Science and Engineering; School of Mathematics and Computational Science, Xiangtan University, Xiangtan, China)

Hailiang Liu (Iowa State University, Mathematics Department, Ames, Ia., U.S.A.)

Nianyu Yi (Hunan Key Laboratory for Computation and Simulation in Science and Engineering; School of Mathematics and Computational Science, Xiangtan University, Xiangtan, China)

Abstract

In this work, we design, analyze and test a conservative discontinuous Galerkin method for solving the Degasperis-Procesi equation. This model is integrable and admits possibly discontinuous solutions, and therefore suitable for modeling both short wave breaking and long wave propagation phenomena. The proposed numerical method is high order accurate, and preserves two invariants, mass and energy, of this nonlinear equation, hence producing wave solutions with satisfying long time behavior. The $L^2$-stability of the scheme for general solutions is a consequence of the energy preserving property. 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 method.

Keywords

discontinuous Galerkin method, Degasperis-Procesi equation, conservation, stability

2010 Mathematics Subject Classification

35Q53, 65M12, 65M60

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Published 29 April 2014