### Abstract

The sine-Gordon equation is one of the basic equations in modern nonlinear wave theory. It has applications in many areas of physics and mathematics. In this paper, we develop and analyze an energy-conserving local discontinuous Galerkin (LDG) method for the two-dimensional sine-Gordon nonlinear hyperbolic equation on Cartesian grids. We prove the energy conserving property, the L
^{2}
stability, and optimal L
^{2}
error estimates for the semi-discrete method. More precisely, we identify special numerical fluxes and a suitable projection of the initial conditions for the LDG scheme to achieve p + 1 order of convergence for both the potential and its gradient in the L
^{2}
-norm, when tensor product polynomials of degree at most p are used. We present several numerical examples to validate the theoretical results. Our numerical examples show the sharpness of the O(h
^{p}
+1) estimate.

Original language | English (US) |
---|---|

Pages (from-to) | 436-462 |

Number of pages | 27 |

Journal | International Journal of Numerical Analysis and Modeling |

Volume | 16 |

Issue number | 3 |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- A priori error estimates
- Cartesian grids
- Energy conservation
- L stability
- Local discontinuous galerkin method
- Sine-Gordon equation

### ASJC Scopus subject areas

- Numerical Analysis

### Cite this

**Optimal error estimates of the local discontinuous galerkin method for the two-dimensional sine-gordon equation on cartesian grids.** / Baccouch, Mahboub.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Optimal error estimates of the local discontinuous galerkin method for the two-dimensional sine-gordon equation on cartesian grids

AU - Baccouch, Mahboub

PY - 2019/1/1

Y1 - 2019/1/1

N2 - The sine-Gordon equation is one of the basic equations in modern nonlinear wave theory. It has applications in many areas of physics and mathematics. In this paper, we develop and analyze an energy-conserving local discontinuous Galerkin (LDG) method for the two-dimensional sine-Gordon nonlinear hyperbolic equation on Cartesian grids. We prove the energy conserving property, the L 2 stability, and optimal L 2 error estimates for the semi-discrete method. More precisely, we identify special numerical fluxes and a suitable projection of the initial conditions for the LDG scheme to achieve p + 1 order of convergence for both the potential and its gradient in the L 2 -norm, when tensor product polynomials of degree at most p are used. We present several numerical examples to validate the theoretical results. Our numerical examples show the sharpness of the O(h p +1) estimate.

AB - The sine-Gordon equation is one of the basic equations in modern nonlinear wave theory. It has applications in many areas of physics and mathematics. In this paper, we develop and analyze an energy-conserving local discontinuous Galerkin (LDG) method for the two-dimensional sine-Gordon nonlinear hyperbolic equation on Cartesian grids. We prove the energy conserving property, the L 2 stability, and optimal L 2 error estimates for the semi-discrete method. More precisely, we identify special numerical fluxes and a suitable projection of the initial conditions for the LDG scheme to achieve p + 1 order of convergence for both the potential and its gradient in the L 2 -norm, when tensor product polynomials of degree at most p are used. We present several numerical examples to validate the theoretical results. Our numerical examples show the sharpness of the O(h p +1) estimate.

KW - A priori error estimates

KW - Cartesian grids

KW - Energy conservation

KW - L stability

KW - Local discontinuous galerkin method

KW - Sine-Gordon equation

UR - http://www.scopus.com/inward/record.url?scp=85063491939&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063491939&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85063491939

VL - 16

SP - 436

EP - 462

JO - International Journal of Numerical Analysis and Modeling

JF - International Journal of Numerical Analysis and Modeling

SN - 1705-5105

IS - 3

ER -