Discontinuous Galerkin methods for short pulse type equations via hodograph transformations

07/18/2019
by   Qian Zhang, et al.
0

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. Thus, 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 adopted 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
08/11/2019

Discontinuous Galerkin methods for the Ostrovsky-Vakhnenko equation

In this paper, we develop discontinuous Galerkin (DG) methods for the Os...
research
08/30/2022

Loop dynamics of a fully discrete short pulse equation

In this article, a fully discrete short pulse (SP) equation is presented...
research
05/02/2023

Discontinuous Galerkin Methods with Generalized Numerical Fluxes for the Vlasov-Viscous Burgers' System

In this paper, semi-discrete numerical scheme for the approximation of t...
research
03/26/2021

Dissipative and conservative local discontinuous Galerkin methods for the Fornberg-Whitham type equations

In this paper, we construct high order energy dissipative and conservati...
research
11/08/2022

Finding twin smooth integers by solving Pell equations

Any pair of consecutive B-smooth integers for a given smoothness bound B...
research
10/08/2021

Local discontinuous Galerkin methods for the abcd nonlinear Boussinesq system

Boussinesq type equations have been widely studied to model the surface ...
research
09/01/2020

HJB and Fokker-Planck equations for river environmental management based on stochastic impulse control with discrete and random observation

We formulate a new two-variable river environmental restoration problem ...

Please sign up or login with your details

Forgot password? Click here to reset