# Optimal energy-conserving local discontinuous Galerkin method for the one-dimensional sine-Gordon equation

Research output: Contribution to journalArticle

6 Citations (Scopus)

### Abstract

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose and analyze a high-order and energy-conserving local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equation in one space dimension. We prove the energy-conserving property and the L2 stability for the semi-discrete LDG method. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L2 -norm for the semi-discrete formulation. In particular, we identify a special numerical flux and a particular projection of the initial conditions for the LDG scheme for which the L2 -norm of the solution and its spatial derivative are of order p+1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal order of convergence. Several numerical results are presented to validate the theoretical analyze of the proposed algorithm. It appears that similar conclusions are valid for the two-dimensional case.

Original language English (US) 316-344 29 International Journal of Computer Mathematics 94 2 https://doi.org/10.1080/00207160.2015.1105364 Published - Feb 1 2017

### Fingerprint

sine-Gordon equation
Local Discontinuous Galerkin Method
Sine-Gordon Equation
Galerkin methods
Derivatives
Nonlinear Hyperbolic Equation
Norm
Derivative
A Priori Error Estimates
Optimal Error Estimates
Discontinuous Galerkin
Auxiliary Variables
Piecewise Polynomials
Convergence of numerical methods
Order of Convergence
Energy
High Energy
Nonlinear Equations
Initial conditions
Numerical Experiment

### Keywords

• Sine-Gordon equation
• a priori error estimates
• energy conservation
• local discontinuous Galerkin method
• stability

### ASJC Scopus subject areas

• Computer Science Applications
• Computational Theory and Mathematics
• Applied Mathematics

### Cite this

In: International Journal of Computer Mathematics, Vol. 94, No. 2, 01.02.2017, p. 316-344.

Research output: Contribution to journalArticle

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N2 - The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose and analyze a high-order and energy-conserving local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equation in one space dimension. We prove the energy-conserving property and the L2 stability for the semi-discrete LDG method. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L2 -norm for the semi-discrete formulation. In particular, we identify a special numerical flux and a particular projection of the initial conditions for the LDG scheme for which the L2 -norm of the solution and its spatial derivative are of order p+1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal order of convergence. Several numerical results are presented to validate the theoretical analyze of the proposed algorithm. It appears that similar conclusions are valid for the two-dimensional case.

AB - The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose and analyze a high-order and energy-conserving local discontinuous Galerkin (LDG) method for the sine-Gordon nonlinear hyperbolic equation in one space dimension. We prove the energy-conserving property and the L2 stability for the semi-discrete LDG method. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L2 -norm for the semi-discrete formulation. In particular, we identify a special numerical flux and a particular projection of the initial conditions for the LDG scheme for which the L2 -norm of the solution and its spatial derivative are of order p+1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal order of convergence. Several numerical results are presented to validate the theoretical analyze of the proposed algorithm. It appears that similar conclusions are valid for the two-dimensional case.

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