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Finite Element $θ$-Schemes for the Acoustic Wave Equation

Finite Element $θ$-Schemes for the Acoustic Wave Equation
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Year:    2011

Author:    Samir Karaa

Advances in Applied Mathematics and Mechanics, Vol. 3 (2011), Iss. 2 : pp. 181–203

Abstract

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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.10-m1018

Advances in Applied Mathematics and Mechanics, Vol. 3 (2011), Iss. 2 : pp. 181–203

Published online:    2011-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    23

Keywords:    Finite element methods discontinuous Galerkin methods wave equation implicit methods energy method stability condition optimal error estimates.

Author Details

Samir Karaa

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