Finite Element θ-Schemes for the Acoustic Wave Equation

2011 ◽  
Vol 3 (1) ◽  
pp. 181-203 ◽  
Author(s):  
Samir Karaa
Keyword(s):  
Finite Element ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Stability And Convergence ◽  
Optimal Error Estimates ◽  
Finite Difference Schemes ◽  
Time Interval ◽  
Acoustic Wave Equation ◽  
Time Step ◽  
Polynomial Degree

AbstractIn 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 L2-norm error over a finite time interval converges optimally as O(hp+1 + ∆ts), where p denotes the polynomial degree, s=2 or 4, h the mesh size, and ∆t the time step.

Error Analysis of Chebyshev Spectral Element Methods for the Acoustic Wave Equation in Heterogeneous Media

2018 ◽  
Vol 26 (03) ◽  
pp. 1850035 ◽  
Author(s):  
Saulo Pomponet Oliveira
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Error Analysis ◽  
Heterogeneous Media ◽  
Spectral Element ◽  
Variable Density ◽  
Dirichlet Boundary Conditions ◽  
Acoustic Wave Equation ◽  
Polynomial Degree ◽  
Mesh Parameter

This work concerns the error analysis of the spectral element method with Gauss–Lobatto–Chebyshev collocation points with the implicit Newmark average acceleration scheme for the two-dimensional acoustic wave equation. The analysis is restricted to homogeneous Dirichlet boundary conditions, constant compressibility and variable density. The proposed error estimates are optimal with respect to the mesh parameter although suboptimal on the polynomial degree. Numerical examples illustrate the theoretical results.

ADAPTIVE FINITE ELEMENT TECHNIQUES FOR THE ACOUSTIC WAVE EQUATION

2001 ◽  
Vol 09 (02) ◽  
pp. 575-591 ◽  
Author(s):  
WOLFGANG BANGERTH ◽  
ROLF RANNACHER
Keyword(s):  
Finite Element ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Efficient Computation ◽  
Acoustic Wave Equation ◽  
Grid Adaptation ◽  
Local Cell ◽  
Error Indicators

We present an adaptive finite element method for solving the acoustic wave equation. Using a global duality argument and Galerkin orthogonality, we derive an identity for the error with respect to an arbitrary functional output of the solution. The error identity is evaluated by solving the dual problem numerically. The resulting local cell-wise error indicators are used in the grid adaptation process. In this way, the space-time mesh can be tailored for the efficient computation of the quantity of interest. We give an overview of the implementation of the proposed method and illustrate its performance by several numerical examples.

Modeling of the acoustic wave equation with transform methods

Geophysics ◽  
1981 ◽  
Vol 46 (6) ◽  
pp. 854-859 ◽  
Author(s):  
Jenö Gazdag
Keyword(s):  
Wave Equation ◽  
Numerical Methods ◽  
Acoustic Wave ◽  
Time Discretization ◽  
Acoustic Wave Equation ◽  
Multiple Reflections ◽  
Time Step ◽  
Transform Methods ◽  
Truncation Errors ◽  
Wave Phenomena

Numerical methods are described for the simulation of wave phenomena with application to the modeling of seismic data. Two separate topics are studied. The first deals with the solution of the acoustic wave equation. The second topic treats wave phenomena whose direction of propagation is restricted within ±90 degrees from a given axis. In the numerical methods developed here, the wave field is advanced in time by using standard time differencing schemes. On the other hand, expressions including space derivative terms are computed by Fourier transform methods. This approach to computing derivatives minimizes truncation errors. Another benefit of transform methods becomes evident when attempting to restrict propagation to upward moving waves, e.g., to avoid multiple reflections. Constraints imposed on the direction of the wave propagation are accomplished most precisely in the wavenumber domain. The error analysis of the algorithms shows that truncation errors are due mainly to time discretization. Such errors can be limited by the choice of the time step. Perhaps the most significant error phenomenon is related to aliasing. This becomes noticeable when a narrow pulse traverses a region with strong velocity variations. It is shown that aliasing errors can be limited by the choice of the pulse width. The feasibility of these modeling methods is demonstrated on numerical examples.

scholarly journals An Unconditionally Stable Method for Solving the Acoustic Wave Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Zhi-Kai Fu ◽  
Li-Hua Shi ◽  
Zheng-Yu Huang ◽  
Shang-Chen Fu
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Testing Procedure ◽  
Basis Functions ◽  
Acoustic Wave Equation ◽  
Stable Method ◽  
Time Step ◽  
Orthogonal Functions ◽  
Unconditionally Stable ◽  
The Time Domain

An unconditionally stable method for solving the time-domain acoustic wave equation using Associated Hermit orthogonal functions is proposed. The second-order time derivatives in acoustic wave equation are expanded by these orthogonal basis functions. By applying Galerkin temporal testing procedure, the time variable can be eliminated from the calculations. The restriction of Courant-Friedrichs-Levy (CFL) condition in selecting time step for analyzing thin layer can be avoided. Numerical results show the accuracy and the efficiency of the proposed method.

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