Paper Info
Title
Discontinuous Galerkin methods for short pulse type equations via hodograph transformations.
Abstract
In the present paper, we consider the discontinuous Galerkin (DG) methods for solving short pulse (SP) type equations. The short pulse equation has been shown to be completely integrable, which admits the loop-soliton, cuspon-soliton solutions as well as smooth-soliton solutions. Through hodograph transformations, these nonclassical solutions can be profiled as the smooth solutions of the coupled dispersionless (CD) system or the sine-Gordon equation. Therefore, DG methods can be developed for the CD system or the sine-Gordon equation to simulate the loop-soliton or cuspon-soliton solutions of the SP equation. The conservativeness or dissipation of the Hamiltonian or momentum for the semi-discrete DG schemes can be proved. Also we modify the above DG schemes and obtain an integration DG scheme. Theoretically the a-priori error estimates have been provided for the momentum conserved DG scheme and the integration DG scheme. We also propose the DG scheme and the integration DG scheme for the sine-Gordon equation, in case the SP equation can not be transformed to the CD system. All these DG schemes can be applied to the generalized or modified SP type equations. Numerical experiments are provided to illustrate the optimal order of accuracy and capability of these DG schemes.
Year
DOI
Venue
2019
10.1016/j.jcp.2019.108928
Journal of Computational Physics
Keywords
Field
DocType
Discontinuous Galerkin method,Short pulse equation,Nonclassical soliton solution,Conservative scheme,Hodograph transformation
Integrable system,Discontinuous Galerkin method,Order of accuracy,Hamiltonian (quantum mechanics),Dissipation,Mathematical analysis,Pulse (signal processing),Momentum,Hodograph,Mathematics
Journal
Volume
ISSN
Citations 
399
0021-9991
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Qian Zhang100.34
Yinhua Xia29710.49