@Article{AAMM-3-2,
author = {Samir, Karaa},
title = {Finite Element $θ$-Schemes for the Acoustic Wave Equation},
journal = {Advances in Applied Mathematics and Mechanics},
year = {2011},
volume = {3},
number = {2},
pages = {181--203},
abstract = {<p style="text-align: justify;">In this paper, we investigate the stability and convergence of a family of
implicit finite difference schemes in time and Galerkin finite element methods in
space for the numerical solution of the acoustic wave equation. The schemes cover
the classical explicit second-order leapfrog scheme and the fourth-order accurate
scheme in time obtained by the modified equation method. We derive general stability
conditions for the family of implicit schemes covering some well-known CFL
conditions. Optimal error estimates are obtained. For sufficiently smooth solutions,
we demonstrate that the maximal error in the $L^2$-norm error over a finite time interval
converges optimally as $\mathcal{O}(h^{p+1}+∆t^s)$, where $p$ denotes the polynomial degree, $s$=2 or 4, $h$ the mesh size, and $∆t$ the time step.</p>},
issn = {2075-1354},
doi = {https://doi.org/10.4208/aamm.10-m1018},
url = {https://global-sci.com/article/73588/finite-element-th-schemes-for-the-acoustic-wave-equation}
}