### 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 O(h ^{p+1} Δ+t ^{s}), where p denotes the polynomial degree, s=2 or 4, h the mesh size, and Δt the time step.

Original language | English |
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Pages (from-to) | 181-203 |

Number of pages | 23 |

Journal | Advances in Applied Mathematics and Mechanics |

Volume | 3 |

Issue number | 2 |

Publication status | Published - 2011 |

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### Keywords

- Discontinuous galerkin methods
- Energy method
- Finite element methods
- Implicit methods
- Optimal error estimates
- Stability condition
- Wave equation

### ASJC Scopus subject areas

- Applied Mathematics
- Mechanical Engineering

### Cite this

**Finite element θ-schemes for the acoustic wave equation.** / Karaa, Samir.

Research output: Contribution to journal › Article

*Advances in Applied Mathematics and Mechanics*, vol. 3, no. 2, pp. 181-203.

}

TY - JOUR

T1 - Finite element θ-schemes for the acoustic wave equation

AU - Karaa, Samir

PY - 2011

Y1 - 2011

N2 - 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 O(h p+1 Δ+t s), where p denotes the polynomial degree, s=2 or 4, h the mesh size, and Δt the time step.

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 O(h p+1 Δ+t s), where p denotes the polynomial degree, s=2 or 4, h the mesh size, and Δt the time step.

KW - Discontinuous galerkin methods

KW - Energy method

KW - Finite element methods

KW - Implicit methods

KW - Optimal error estimates

KW - Stability condition

KW - Wave equation

UR - http://www.scopus.com/inward/record.url?scp=79957585404&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79957585404&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:79957585404

VL - 3

SP - 181

EP - 203

JO - Advances in Applied Mathematics and Mechanics

JF - Advances in Applied Mathematics and Mechanics

SN - 2070-0733

IS - 2

ER -