scholarly journals On the Accuracy and Efficiency of Transient Spectral Element Models for Seismic Wave Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Sanna Mönkölä
Keyword(s):  
Fourth Order ◽  
Wave Equations ◽  
Seismic Waves ◽  
Time Discretization ◽  
Spectral Element ◽  
Higher Order ◽  
Time Stepping ◽  
Discretization Schemes ◽  
Elastic Wave Equations ◽  
Wave Problems

This study concentrates on transient multiphysical wave problems for simulating seismic waves. The presented models cover the coupling between elastic wave equations in solid structures and acoustic wave equations in fluids. We focus especially on the accuracy and efficiency of the numerical solution based on higher-order discretizations. The spatial discretization is performed by the spectral element method. For time discretization we compare three different schemes. The efficiency of the higher-order time discretization schemes depends on several factors which we discuss by presenting numerical experiments with the fourth-order Runge-Kutta and the fourth-order Adams-Bashforth time-stepping. We generate a synthetic seismogram and demonstrate its function by a numerical simulation.

Numerical simulation of higher-order diffusion-wave equations of variable coefficients using the meshless spectral method

2020 ◽  
pp. 2150010
Author(s):  
Manzoor Hussain ◽  
Sirajul Haq
Keyword(s):  
Wave Equations ◽  
Interpolation Method ◽  
Variable Coefficients ◽  
Higher Order ◽  
Implicit Time ◽  
Test Problems ◽  
Time Step ◽  
Diffusion Wave ◽  
Time Step Size ◽  
Time Stepping

In this paper, meshless spectral interpolation technique using implicit time stepping scheme is proposed for the numerical simulations of time-fractional higher-order diffusion wave equations (TFHODWEs) of variable coefficients. Meshless shape functions, obtained from radial basis functions (RBFs) and point interpolation method (PIM), are used for spatial approximation. Central differences coupled with quadrature rule of [Formula: see text] are employed for fractional temporal approximation. For advancement of solution, an implicit time stepping scheme is then invoked. Simulations performed for different benchmark test problems feature good agreement with exact solutions. Stability analysis of the proposed method is theoretically discussed and computationally validated to support the analysis. Accuracy and efficiency of the proposed method are assessed via [Formula: see text], [Formula: see text] and [Formula: see text] error norms as well as number of nodes [Formula: see text] and time step-size [Formula: see text].

Modelling the effects of diffusive-viscous waves in a 3-D fluid-saturated media using two numerical approaches

2020 ◽  
Vol 224 (2) ◽  
pp. 1443-1463
Author(s):  
Victor Mensah ◽  
Arturo Hidalgo
Keyword(s):  
Seismic Wave ◽  
Fourth Order ◽  
Seismic Waves ◽  
Time Discretization ◽  
Seismic Wave Propagation ◽  
Second Order ◽  
Finite Volume Scheme ◽  
Fluid Saturation ◽  
Runge Kutta ◽  
Saturated Medium

SUMMARY The accurate numerical modelling of 3-D seismic wave propagation is essential in understanding details to seismic wavefields which are, observed on regional and global scales on the Earth’s surface. The diffusive-viscous wave (DVW) equation was proposed to study the connection between fluid saturation and frequency dependence of reflections and to characterize the attenuation property of the seismic wave in a fluid-saturated medium. The attenuation of DVW is primarily described by the active attenuation parameters (AAP) in the equation. It is, therefore, imperative to acquire these parameters and to additionally specify the characteristics of the DVW. In this paper, quality factor, Q is used to obtain the AAP, and they are compared to those of the visco-acoustic wave. We further derive the 3-D numerical schemes based on a second order accurate finite-volume scheme with a second order Runge–Kutta approximation for the time discretization and a fourth order accurate finite-difference scheme with a fourth order Runge–Kutta approximation for the time discretization. We then simulate the propagation of seismic waves in a 3-D fluid-saturated medium based on the derived schemes. The numerical results indicate stronger attenuation when compared to the visco-acoustic case.

The spectral element method for elastic wave equations—application to 2-D and 3-D seismic problems

1999 ◽  
Vol 45 (9) ◽  
pp. 1139-1164 ◽  
Author(s):  
Dimitri Komatitsch ◽  
Jean-Pierre Vilotte ◽  
Rossana Vai ◽  
José M. Castillo-Covarrubias ◽  
Francisco J. Sánchez-Sesma
Keyword(s):  
Elastic Wave ◽  
Wave Equations ◽  
Spectral Element Method ◽  
Spectral Element ◽  
Elastic Wave Equations ◽  
Element Method

Numerical Solution of the Nonlinear Wave Equation via Fourth-Order Time Stepping

2015 ◽  
Vol 729 ◽  
pp. 213-219
Author(s):  
Mohammadreza Askaripour Lahiji ◽  
Zainal Abdul Aziz
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Fourth Order ◽  
Nonlinear Wave Equation ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Fisher Equation ◽  
Exponential Time ◽  
Exponential Time Differencing ◽  
Time Stepping

Some nonlinear wave equations are more difficult to solve analytically. Exponential Time Differencing (ETD) technique requires minimum stages to obtain the required accurateness, which suggests an efficient technique relating to computational duration that ensures remarkable stability characteristics upon resolving the nonlinear wave equations. This article solves the non-diagonal example of Fisher equation via the exponential time differencing Runge-Kutta 4 method (ETDRK4). Implementation of the method is demonstrated by short Matlab programs.

A new high-order scheme based on numerical dispersion analysis of the wave phase velocity for semidiscrete wave equations

Geophysics ◽  
2018 ◽  
Vol 83 (3) ◽  
pp. T123-T138 ◽  
Author(s):  
Xiao Ma ◽  
Dinghui Yang ◽  
Xijun He ◽  
Jingshuang Li ◽  
Yongchang Zheng
Keyword(s):  
Differential Operator ◽  
Acoustic Wave ◽  
Phase Velocity ◽  
Wave Equations ◽  
Dispersion Analysis ◽  
Numerical Dispersion ◽  
Spatial Discretization ◽  
Fully Discrete ◽  
Time Stepping ◽  
Discretization Schemes

In the numerical computation of wave equations, numerical dispersion is a persistent problem arising from inadequate discretization of the continuous wave equation. To thoroughly understand the mechanism of numerical dispersion, we separately analyze the numerical dispersion relations of time-stepping and spatial discretization schemes by Fourier analysis. The relevant results show that the numerical dispersion errors of time-marching schemes depend on the time step length or the Courant number, whereas the numerical dispersion errors of spatial discretization schemes are determined by the error between the eigenvalues of the numerical spatial differential operator and the continuous spatial differential operator. We also find that the much better numerical dispersion accuracy of the stereo-modeling discrete (SMD)-type operator can be attributed to the inclusion of diversified basis functions for its eigenvalue. Based on these findings, we combine the optimal four-stage symplectic partitioned Runge-Kutta and eighth-order SMD as a new fully discrete scheme. The subsequent analysis of its normalized phase velocity is consistent with the numerical dispersion analysis in semidiscrete forms. This is followed by an acoustic wave simulation in a homogeneous model and corresponding computational efficiency comparison. The results show that the new scheme is much more accurate and antidispersive on a coarse grid. In the final two numerical experiments, we use the new scheme to model the acoustic wave propagation in a three-layer model and the Marmousi model. The convolutional perfectly matched layer is applied to eliminate artificial boundary reflections. Our semidiscrete numerical dispersion analysis provides an efficient tool to quantitatively evaluate the time-stepping and spatial discretization schemes. It can facilitate the development of more accurate fully discrete numerical schemes for solving seismic wave equations.

TACHYONS AND HIGHER ORDER WAVE EQUATIONS

1995 ◽  
Vol 10 (12) ◽  
pp. 1737-1747 ◽  
Author(s):  
D.G. BARCI ◽  
C.G. BOLLINI ◽  
M.C. ROCCA
Keyword(s):  
Wave Equation ◽  
Green Function ◽  
Fourth Order ◽  
Wave Equations ◽  
Higher Order ◽  
Causal Function ◽  
Fourth Order Wave Equation

We consider a fourth order wave equation having normal as well as tachyonic solutions. The propagators are, respectively, the Feynman causal function and the Wheeler-Green function (half advanced and half retarded). The latter is consistent with the elimination of tachyons from free asymptotic states. We verify the absence of absorptive parts from convolutions involving the tachyon propagator.

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