Error analysis of the spectral element method with Gauss–Lobatto–Legendre points for the acoustic wave equation in heterogeneous media

2018 ◽  
Vol 129 ◽  
pp. 39-57 ◽  
Author(s):  
Saulo Pomponet Oliveira ◽  
Stela Angelozi Leite
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Error Analysis ◽  
Spectral Element Method ◽  
Heterogeneous Media ◽  
Spectral Element ◽  
Acoustic Wave Equation ◽  
Element Method

scholarly journals Space-time spectral element method solution for the acoustic wave equation and its dispersion analysis

2017 ◽  
Vol 38 (6) ◽  
pp. 303-313 ◽  
Author(s):  
Yanhui Geng ◽  
Guoliang Qin ◽  
Jiazhong Zhang ◽  
Wenqiang He ◽  
Zhenzhong Bao ◽  
...  
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Spectral Element Method ◽  
Spectral Element ◽  
Dispersion Analysis ◽  
Space Time ◽  
Acoustic Wave Equation ◽  
Element Method

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.

An analytic signal based accurate time-domain visco-acoustic wave equation from the Constant-Q theory

Geophysics ◽  
2021 ◽  
pp. 1-58
Author(s):  
Hongwei Liu ◽  
Yi Luo
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Time Domain ◽  
Seismic Waves ◽  
Heterogeneous Media ◽  
Relaxation Property ◽  
Analytic Signal ◽  
Acoustic Wave Equation ◽  
Real Component ◽  
Frequency Components

We present a concise time-domain wave equation to accurately simulate wave propagation in visco-acoustic media. The central idea behind this work is to dismiss the negative frequency components from a time-domain signal by converting the signal to its analytic format. The negative frequency components of any analytic signal are always zero, meaning we can construct the visco-acoustic wave equation to honor the relaxation property of the media for positive frequencies only. The newly proposed complex-valued wave equation (CWE) represents the wavefield with its analytic signal, whose real part is the desired physical wavefield, while the imaginary part is the Hilbert transform of the real component. Specifically, this CWE is accurate for both weak and strong attenuating media in terms of both dissipation and dispersion and the attenuation is precisely linear with respect to the frequencies. Besides, the CWE is easy and flexible to model dispersion-only, dissipation-only or dispersion-plus-dissipation seismic waves. We have verified these CWEs by comparing the results with analytical solutions, and achieved nearly perfect matching. Except for the homogeneous Q media, we have also extended the CWEs to heterogeneous media. The results of the CWEs for heterogeneous Q media are consistent with those computed from the nonstationary operator based Fourier Integral method and from the Standard Linear Solid (SLS) equations.

scholarly journals Efficiency of the spectral element method with very high polynomial degree to solve the elastic wave equation

Geophysics ◽  
2020 ◽  
Vol 85 (1) ◽  
pp. T33-T43
Author(s):  
Chao Lyu ◽  
Yann Capdeville ◽  
Liang Zhao
Keyword(s):  
Wave Equation ◽  
Spectral Element Method ◽  
Computing Time ◽  
Spectral Element ◽  
Two Dimensions ◽  
Elastic Model ◽  
Time Step ◽  
High Degree ◽  
Very High ◽  
Element Method

The spectral element method (SEM) has gained tremendous popularity within the seismological community to solve the wave equation at all scales. Classic SEM applications mostly rely on degrees 4–8 elements in each tensorial direction. Higher degrees are usually not considered due to two main reasons. First, high degrees imply large elements, which make the meshing of mechanical discontinuities difficult. Second, the SEM’s collocation points cluster toward the edge of the elements with the degree, degrading the time-marching stability criteria and imposing a small time step and a high numerical cost. Recently, the homogenization method has been introduced in seismology. This method can be seen as a preprocessing step before solving the wave equation that smooths out the internal mechanical discontinuities of the elastic model. It releases the meshing constraint and makes use of very high degree elements more attractive. Thus, we address the question of memory and computing time efficiency of very high degree elements in SEM, up to degree 40. Numerical analyses reveal that, for a fixed accuracy, very high degree elements require less computer memory than low-degree elements. With minimum sampling points per minimum wavelength of 2.5, the memory needed for a degree 20 is about a quarter that of the one necessary for a degree 4 in two dimensions and about one-eighth in three dimensions. Moreover, for the SEM codes tested in this work, the computation time with degrees 12–24 can be up to twice faster than the classic degree 4. This makes SEM with very high degrees attractive and competitive for solving the wave equation in many situations.

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