Volume 3, Issue 2
Finite Element $θ$-Schemes for the Acoustic Wave Equation

Adv. Appl. Math. Mech., 3 (2011), pp. 181-203.

Published online: 2011-03

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• 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.

• Keywords

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

65M60, 65M12, 65M15

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• RIS
• TXT
@Article{AAMM-3-181, 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 = {

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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.10-m1018}, url = {http://global-sci.org/intro/article_detail/aamm/164.html} }
TY - JOUR T1 - Finite Element $θ$-Schemes for the Acoustic Wave Equation AU - Samir Karaa , JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 181 EP - 203 PY - 2011 DA - 2011/03 SN - 3 DO - http://doi.org/10.4208/aamm.10-m1018 UR - https://global-sci.org/intro/article_detail/aamm/164.html KW - Finite element methods, discontinuous Galerkin methods, wave equation, implicit methods, energy method, stability condition, optimal error estimates. AB -

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.

Samir Karaa. (1970). Finite Element $θ$-Schemes for the Acoustic Wave Equation. Advances in Applied Mathematics and Mechanics. 3 (2). 181-203. doi:10.4208/aamm.10-m1018
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