@Article{IJNAM-16-3,
author = {Mahboub, Baccouch},
title = {Optimal Error Estimates of the Local Discontinuous Galerkin Method for the Two-Dimensional Sine-Gordon Equation on Cartesian Grids},
journal = {International Journal of Numerical Analysis and Modeling},
year = {2019},
volume = {16},
number = {3},
pages = {436--462},
abstract = {<p style="text-align: justify;">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 $L^2$<sup>&nbsp;</sup>stability, and optimal $L^2$&nbsp;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 $\mathcal{O}$($h^{p+1}$) estimate.</p>},
issn = {2617-8710},
doi = {https://doi.org/2019-IJNAM-12877},
url = {https://global-sci.com/article/83072/optimal-error-estimates-of-the-local-discontinuous-galerkin-method-for-the-two-dimensional-sine-gordon-equation-on-cartesian-grids}
}