Optimal blended spectral-element operators for acoustic wave modeling

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. SM95-SM106 ◽  
Author(s):  
Géza Seriani ◽  
Saulo P. Oliveira
Keyword(s):  
Wave Equations ◽  
Mass Matrix ◽  
Spectral Element ◽  
Spatial Dimension ◽  
Dispersion Analysis ◽  
High Order ◽  
Numerical Dispersion ◽  
Product Decomposition ◽  
Wave Operators ◽  
Very High

Spectral-element methods, based on high-order polynomials, are among the most commonly used techniques for computing accurate simulations of wave propagation phenomena in complex media. However, to retain computational efficiency, very high order polynomials cannot be used and errors such as numerical dispersion and numerical anisotropy cannot be totally avoided. In the present work, we devise an approach for reducing such errors by considering modified discrete wave operators. We analyze consistent and lumped operators together with blended operators (weighted averages of consistent and lumped operators). Furthermore, using the operator-blending approach and a novel dispersion analysis method, we develop optimal spectral-element operators that have increased numerical accuracy, without resorting to very high order operators. The new operators are faster and computationally more efficient than consistent operators. Our approach is based on the tensor product decomposition of the element matrices into 1D factors. We apply standard lumping to the factor associated with the 1D mass matrix. A simplified numerical dispersion analysis of arbitrary order and spatial dimension provides a practical criterion for weighting consistent and lumped matrices. The approach is general and is valid for solving both the time-dependent and the stationary (Helmholtz) wave equations.

scholarly journals Dispersion analysis of the discontinuous Galerkin method as applied to the equations of dynamic elasticity theory

2015 ◽  
pp. 387-406
Author(s):  
В.В. Лисица
Keyword(s):  
Galerkin Method ◽  
Elasticity Theory ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Computational Cost ◽  
Dispersion Analysis ◽  
Numerical Dispersion ◽  
Seismic Modeling ◽  
Dynamic Elasticity ◽  
Very High

Приводится дисперсионный анализ разрывного метода Галеркина в применении к системе уравнений динамической теории упругости. В зависимости от степени базисных полиномов рассматриваются P1-, P2- и P3-формулировки метода при использовании регулярной треугольной сетки. Показано, что для задач сейсмического моделирования оптимальной является P2-формулировка, поскольку сочетает в себе достаточную точность (численная дисперсия не выше 0.05% и вычислительную эффективность. Использование P1-формулировки приводит к недопустимо высокой численной дисперсии, в то время как P3-формулировка является чрезвычайно ресурсоемкой при использовании дискретизаций от 3 до 20 ячеек сетки на длину волны, типичной для сейсмического моделирования. The dispersion analysis of the discontinuous Galerkin method as applied to the equations of dynamic elasticity theory is performed. Depending on the degrees of basis polynomials, we consider the P1, P2, and P3 formulations of this method in the case of regular triangular meshes. It is shown that, for the problems of seismic modeling, the P2 formulation is optimal, since a sufficient accuracy (the numerical dispersion does not exceed 0.05%) and the computational efficiency are achieved. The application of the P1 formulation leads to an undesirably high numerical dispersion. The P3 formulation allows one to obtain accurate results, but its computational cost is very high when the number of grid cells per wavelength belongs to range between 3 and 20, which is typical for the seismic modeling.

Electromagnetic wave propagation based upon spectral-element methodology in dispersive and attenuating media

2019 ◽  
Vol 220 (2) ◽  
pp. 951-966
Author(s):  
Christina Morency
Keyword(s):  
Wave Propagation ◽  
Dielectric Permittivity ◽  
Seismic Wave ◽  
Wave Equations ◽  
Mass Matrix ◽  
Time Integration ◽  
Spectral Element ◽  
Seismic Wave Propagation ◽  
Transverse Magnetic ◽  
Integration Schemes

SUMMARY We build on mathematical equivalences between Maxwell’s wave equations for an electromagnetic medium and elastic seismic wave equations. This allows us to readily model Maxwell’s wave propagation in the spectral-element codes SPECFEM2D and SPECFEM3D, written for acoustic, viscoelastic and poroelastic seismic wave propagation, providing the ability to handle complex geometries, inherent to finite-element methods and retaining the strength of exponential convergence and accuracy due to the use of high-degree polynomials to interpolate field functions on the elements, characteristic to spectral-element methods (SEMs). Attenuation and dispersion processes related to the frequency dependence of dielectric permittivity and conductivity are also included using a Zener model, similar to shear attenuation in viscoelastic media or viscous diffusion in poroelastic media, and a Kelvin–Voigt model, respectively. Ability to account for anisotropic media is also discussed. Here, we limit ourselves to certain dielectric permittivity tensor geometries, in order to conserve a diagonal mass matrix after discretization of the system of equations. Doing so, simulation of Maxwell’s wave equations in the radar frequency range based on SEM can be solved using explicit time integration schemes well suited for parallel computation. We validate our formulation with analytical solutions. In 2-D, our implementation allows for the modelling of both a transverse magnetic (TM) mode, suitable for surface based reflection ground penetration radar type of applications, and a transverse electric (TE) mode more suitable for crosshole and vertical radar profiling setups. Two 2-D examples are designed to demonstrated the use of the TM and TE modes. A 3-D example is also presented, which allows for the full TEM solution, different antenna orientations, and out-of-plane variations in material properties.

Lowrank seismic-wave extrapolation on a staggered grid

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. T157-T168 ◽  
Author(s):  
Gang Fang ◽  
Sergey Fomel ◽  
Qizhen Du ◽  
Jingwei Hu
Keyword(s):  
Seismic Wave ◽  
Wave Equations ◽  
Variable Density ◽  
Dispersion Analysis ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Order System ◽  
Constant Density ◽  
Reverse Time ◽  
Time Migration

We evaluated a new spectral method and a new finite-difference (FD) method for seismic-wave extrapolation in time. Using staggered temporal and spatial grids, we derived a wave-extrapolation operator using a lowrank decomposition for a first-order system of wave equations and designed the corresponding FD scheme. The proposed methods extend previously proposed lowrank and lowrank FD wave extrapolation methods from the cases of constant density to those of variable density. Dispersion analysis demonstrated that the proposed methods have high accuracy for a wide wavenumber range and significantly reduce the numerical dispersion. The method of manufactured solutions coupled with mesh refinement was used to verify each method and to compare numerical errors. Tests on 2D synthetic examples demonstrated that the proposed method is highly accurate and stable. The proposed methods can be used for seismic modeling or reverse-time migration.

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.

DFT MODAL ANALYSIS OF SPECTRAL ELEMENT METHODS FOR ACOUSTIC WAVE PROPAGATION

2008 ◽  
Vol 16 (04) ◽  
pp. 531-561 ◽  
Author(s):  
GÉZA SERIANI ◽  
SAULO POMPONET OLIVEIRA
Keyword(s):  
Wave Propagation ◽  
Plane Waves ◽  
Spectral Element ◽  
Numerical Dispersion ◽  
Spectral Elements ◽  
Spectral Element Methods ◽  
Wave Operators ◽  
Safe Level ◽  
Novel Approach ◽  
2D And 3D

Spectral element methods are now widely used for wave propagation simulations. They are appreciated for their high order of accuracy, but are still used on a heuristic basis. In this work we present the numerical dispersion of spectral elements, which allows us to assess their error limits and to devise efficient numerical simulations, particularly for 2D and 3D problems. We propose a novel approach based on a discrete Fourier transform of both the probing plane waves and the discrete wave operators. The underlying dispersion relation is estimated by the Rayleigh quotients of the plane waves with respect to the discrete operator. Together with the Kronecker product properties, this approach delivers numerical dispersion estimates for 1D operators as well as for 2D and 3D operators, and is well suited for spectral element methods, which use nonequidistant collocation points such as Gauss–Lobatto–Chebyshev and Gauss–Lobatto–Legendre points. We illustrate this methodology with dispersion and anisotropy graphs for spectral elements with polynomial degrees from 4 to 12. These graphs confirm the rule of thumb that spectral element methods reach a safe level of accuracy at about four grid points per wavelength.

A nearly analytic discrete method for solving the acoustic-wave equations in the frequency domain

Geophysics ◽  
2017 ◽  
Vol 82 (1) ◽  
pp. T43-T57 ◽  
Author(s):  
Chao Lang ◽  
Ding-Hui Yang
Keyword(s):  
Linear System ◽  
Acoustic Wave ◽  
Frequency Domain ◽  
Wave Equations ◽  
Krylov Subspace ◽  
Coefficient Matrix ◽  
Dispersion Analysis ◽  
Numerical Dispersion ◽  
Numerical Schemes ◽  
Wavefield Simulation

We have developed a nearly analytic discrete (NAD) method to discretize frequency-domain acoustic-wave equations with an absorbing boundary condition. We evaluate in detail the discrete process of wave equations to derive a linear system. The sparse structure and eigenproperties of its coefficient matrix (also called the impedance matrix) were analyzed to reveal the intrinsic difficulty in solving the linear system efficiently. To accelerate the forward-modeling process in the frequency domain, we introduce a class of inexact rotated block triangular preconditioners incorporated with Krylov subspace methods to solve this linear system and test their numerical behaviors by comparing with other two commonly used preconditioners. To this end, we perform wavefield simulation by the NAD method and another two conventional numerical schemes in various media. Numerical dispersion analysis and waveform comparison are also implemented for these numerical schemes. Our results show the superiority of our proposed methods.

VERY HIGH-ORDER SPATIALLY IMPLICIT SCHEMES FOR COMPUTATIONAL ACOUSTICS ON CURVILINEAR MESHES

2001 ◽  
Vol 09 (04) ◽  
pp. 1259-1286 ◽  
Author(s):  
MIGUEL R. VISBAL ◽  
DATTA V. GAITONDE
Keyword(s):  
Three Dimensional ◽  
Radiation Condition ◽  
Higher Order ◽  
High Order ◽  
High Frequencies ◽  
Decomposition Strategies ◽  
The Impact ◽  
Spurious Modes ◽  
Very High ◽  
Computational Strategy

A high-order compact-differencing and filtering algorithm, coupled with the classical fourth-order Runge–Kutta scheme, is developed and implemented to simulate aeroacoustic phenomena on curvilinear geometries. Several issues pertinent to the use of such schemes are addressed. The impact of mesh stretching in the generation of high-frequency spurious modes is examined and the need for a discriminating higher-order filter procedure is established and resolved. The incorporation of these filtering techniques also permits a robust treatment of outflow radiation condition by taking advantage of energy transfer to high-frequencies caused by rapid mesh stretching. For conditions on the scatterer, higher-order one-sided filter treatments are shown to be superior in terms of accuracy and stability compared to standard explicit variations. Computations demonstrate that these algorithmic components are also crucial to the success of interface treatments created in multi-domain and domain-decomposition strategies. For three-dimensional computations, special metric relations are employed to assure the fidelity of the scheme in highly curvilinear meshes. A variety of problems, including several benchmark computations, demonstrate the success of the overall computational strategy.

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