Simulation Of Wave Equation Research Articles

A novel explicit optimized scheme for numerical simulation of elastic-wavefield separation

Abstract Numerical simulation of elastic wave equation helps us better understand the information of underground structures and elastic wave imaging has attracted the widespread attention of researchers. Using elastic wave imaging requires separating the compressional and shear wavefields. Therefore, we develop a novel explicit optimized scheme to simulate the separated elastic wavefield. We construct a kind of 1-norm objective function directly utilizing the dispersion error and employ the simulated annealing algorithm to acquire improved finite-difference operators, whose optimal coefficients can effectively suppress spatial numerical dispersion. Meanwhile, we introduce a rotated staggered-grid (RSG) approach to enhance computational stability. Then, our proposed scheme also called the optimized RSG approach is applied to the elastic wave equations and decoupled elastic wave equations to simulate the decoupled compressional and shear wavefield propagation. Numerical dispersion analysis is consistent with numerical results. The waveform comparison shows that the optimized RSG approach possesses higher accuracy, and several complex models are used to validate the applicability and effectiveness of the presented scheme.

3D numerical simulation of frequency-domain elastic wave equation with a compact second-order scheme

Finite-difference frequency-domain (FDFD) modeling has gained popularity in recent years. However, its main drawback lies in the need to solve large sparse linear systems, which is computationally expensive, especially for 3D elastic wave simulations. The existing elastic wave optimal schemes improve the simulation accuracy at the cost of using large-sized stencils or high-order finite-difference (FD) operators, and most of them only consider equal grid spacing and solid media. To address these issues, we have proposed a compact second-order scheme for simulating 3D elastic wave propagation. Starting from the second-order FD discretization of the first-order velocity-stress expression, the compact discrete scheme of the second-order elastic wave equation is obtained by using a parsimonious staggered-grid strategy to eliminate the stress variables. In this way, the scheme preserves the ability to handle high-contrast velocity and liquid-solid media, and has a more compact and sparse impedance matrix than existing schemes, which helps save computational costs. We also use an antilumped mass strategy to further improve the modeling accuracy. Dispersion analysis indicates that our compact second-order scheme has better accuracy than the classic second-order scheme, and even better accuracy than the average derivative 27-point scheme at large Poisson's ratio. Compared with the existing 3D elastic wave schemes, the compact second-order scheme has advantages in computational cost for the same grid. Finally, we employ these schemes to implement 3D numerical simulations, validating the effectiveness of our proposed scheme.

Wavefield simulation of acoustic VTI wave equation based on adaptive-coefficient finite-difference frequency-domain method

Abstract Many simulation methods have been developed for P-waves in vertically transversely isotropic (VTI) media. These methods are based on the acoustic approximation. The finite-difference frequency-domain (FDFD) method stands out for its ability to simulate multi-shot or narrowband seismic data. It has no temporal dispersion, facilitates attenuation modelling, and enables parallelization. The optimal FDFD method is commonly used to simulate the acoustic VTI wave equation, but it applies the same FDFD coefficients for different frequencies and model velocities, which cannot fully minimize the numerical dispersion error. To enhance its accuracy and effectiveness, we develop an adaptive-coefficient FDFD method specifically for the acoustic VTI wave equation. The FDFD coefficients depend on two factors: the number of wavelengths in each grid and the Thomsen parameters. The dispersion analysis reveals that the proposed FDFD method can achieve a reduction in the necessary number of grid points from 4 to 2.5 compared to the optimal 9-point average derivative method (ADM), while maintaining a maximum dispersion error of 1%. From three numerical examples, the developed FDFD method can obtain more accurate wavefield results than the ADM optimal FDFD method, while taking comparable computational time and memory.

Numerical Simulation and Parallel Computing of Acoustic Wave Equation in Isotropic-Heterogeneous Media

Numerical Simulation and Parallel Computing of Acoustic Wave Equation in Isotropic-Heterogeneous Media

Numerical Simulation of Wave Equation Using the Finite Difference Method with Variable Refined Grid in Complex Near-surface Condition

The horizontal layered refined grid is adopted in the conventional finite difference method (FDM) of variable refined grid for the numerical simulation of wave equation. But, the method cannot well adapt to the characteristics of undulated topography and the variation of velocity structure in near-surface region. To solve this problem, a numerical simulation of wave equation using FDM with undulated variable refined grid is proposed in this paper. The grids are generated according to the undulated topography and the spatial distribution of velocity in the near-surface region from low-velocity to the high-velocity layer. For ensuring the accuracy while taking into account the computational efficiency, the FDM with variable coefficient in above grids is used to discretize the acoustic wave equation. The numerical evaluation and example show that the same accuracy as the method using fine grid can be obtained by the new method. However, its computational time is only 43.6% of the one spent by the fine grid. Furthermore, the new method can also stably adapt to the real complex loess plateau near-surface model in a working area of Ordos Basins.

High frequency attenuation of elastic waves transmitted at an angle through a randomly-fluctuating horizontally-layered slab

This paper is concerned with the modeling of elastic waves traveling at small incidence angles through a randomly-fluctuating horizontally-layered slab, in regimes where the wavelength is small compared to the thickness of the slab. The wave propagation problem is reset in a frame following the coherent front, which propagates in a homogenized medium. This homogenized medium is anisotropic because of the layering, and the equations obtained account explicitly for the coupling of quasi-P and quasi-S waves. The resulting model is governed by a set of coupled stochastic ordinary differential equations that can be approximated numerically very efficiently, and yields in particular estimates of the transmission and reflections coefficients of the slab. The latter compare favorably to the coefficients obtained in a full scale numerical simulation of the (micro-scale) wave equation, for a fraction of the cost.

A numerical method for analysis and simulation of diffusive viscous wave equations with variable coefficients on polygonal meshes

A numerical method for analysis and simulation of diffusive viscous wave equations with variable coefficients on polygonal meshes

Simulation of wave propagation with obstacles: Time invariance operator applied to interference and diffraction

We present here a computational numerical operator, and we name it as Time Invariance Operator (TIO). This operator can add obstacles to the domain of the differential equation that describes a physical phenomenon. After the TIO acts, the wave equation recognizes the introduced points as non-interacting zones without affecting the rest of the domain. Computational physics has been consolidated as an important field of study, especially when connected with the fundamentals of physics. In many cases, simulations are conducted considering the ideal case of a wave in an infinite domain and open space without considering obstacles, barriers, or other aspects of the real world. The results presented in this paper allow us to infer that the TIO is the easiest way to apply the physical domain to wave propagation simulations and successfully recreate wave interaction experiments through computer simulations. Our motivation is to perform wave simulations that interact with obstacles, barriers, single slits, and double slits. We aim to investigate the results obtained in images to determine if the methodology we used to introduce realistic physical characteristics was successful in presenting the expected phenomenology. The simplicity of the TIO’s action in creating locally time-invariant regions over the domain makes it suitable not only for waves but also for equations with transient terms. Heat transfer, mass transfer, computational fluid dynamics, and other time evolution equations can take some benefit from the operator presented in this paper. The TIO ensures local conservation that mimics interaction regions or ensures free space characteristics if it is the case like a 2D tensor of local conservation. The principal result from this paper is the validation of the TIO to impose realistic conditions with minimal modifications over a running code of wave equation simulation originally in free space. The TIO is innovative because it imposes dynamic conditions that mimic realistic interacting zones.

A new data-domain reflection full-waveform inversion method based on common-image-point gathers

The migration velocity analysis (MVA) based on the common-image gathers (CIGs) in the image domain is an effective approach for evaluating the seismic background velocity, a crucial index for the full-waveform inversion (FWI) and migration. Compared with cumbersome conventional MVA methods, differential semblance optimization (DSO) is a variation of image-domain wavefield tomography that avoids interactive manual picking but increases the computation cost tremendously. In recent years, the reflection waveform inversions have been studied in depth to construct the background velocity, matching reflection travel time or waveform in the data domain, which requires expensive Born simulations based on accurate imaging results. This study performs a new reflection full-waveform inversion method in the image domain based on the flatness of arbitrarily selected common-image-point (CIP) gathers and their Born simulations. According to the relationship between imaging and background model, the flatness of CIP gathers under the L1 norm is measured. Afterwards, we construct the objective function to update the background model without manual picking or wave-equation migration in an extended dimension. The inversion accuracy is controlled by balancing the density of the imaging points and the computation capability. The whole procedure is implemented in the frequency domain, so the main computation comes from the LU decomposition in the frequency-domain wave-equation simulation, in which way, the proposed data-domain reflection full-waveform inversion achieves complete automation with lower costs. The objective function analysis shows a better convex, proving the weak dependence on the starting model of this method. Besides, synthetic experiments verify the validity and the field data test demonstrates its effectiveness.

Novel first-order k-space formulations for wave propagation by asymmetrical factorization of space-wavenumber domain wave propagators

Wave-equation simulation based on the k-space method produces nearly dispersion-free wavefields and enhances simulation stability. However, for simulation in heterogeneous media, the conventional first-order k-space method requires many mixed-domain operators, which are the most expensive part of the wave-extrapolation process. We have analyzed and summarized the problem of the conventional k-space method as symmetrical factorization of the wave propagators. Based on this analysis, we develop a novel asymmetrical factorization-based k-space method that can significantly reduce the number of mixed-domain operators without compromising modeling accuracy. By using this method, the number of mixed-domain operators is reduced by half, and thus, the computational cost decreases significantly. Furthermore, we have compared our method to the conventional pseudospectral method. The comparison finds that, at comparable accuracy, our method is more efficient due to its ability to use a larger time step. Acoustic and elastic examples demonstrate the correctness and effectiveness of our method.

Local discontinuous Galerkin methods for diffusive–viscous wave equations

Numerical simulation of seismic wave equations has attracted much attention and plays a significant role in exploration seismology. As one of seismic wave models, the diffusive–viscous wave theory usually describes the attenuation of seismic wave propagating in fluid-saturated medium. In this paper, we focus on the design of numerical methods for the diffusive–viscous wave equations with variable coefficients. We develop a local discontinuous Galerkin (LDG) method, in which numerical fluxes are chosen carefully to maintain stability and accuracy. Moreover, we also prove the optimal error estimates for both the energy norm and the L2 norm. Numerical experiments are provided to demonstrate the optimal convergence rate and effectiveness of the proposed LDG method.

A hybrid explicit implicit staggered grid finite-difference scheme for the first-order acoustic wave equation modeling

Implicit staggered-grid finite-difference (SGFD) methods are widely used for the first-order acoustic wave-equation modeling. The identical implicit SGFD operator is commonly used for all of the first-order spatial derivatives in the first-order acoustic wave-equation. In this paper, we propose a hybrid explicit implicit SGFD (HEI-SGFD) scheme which could simultaneously preserve the wave-equation simulation accuracy and increase the wave-equation simulation speed. We use a second-order explicit SGFD operator for half of the first-order spatial derivatives in the first-order acoustic wave-equation. At the same time, we use the implicit SGFD operator with added points in the diagonal direction for the other first-order spatial derivatives in the first-order acoustic wave-equation. The proposed HEI-SGFD scheme nearly doubles the wave-equation simulation speed compared to the implicit SGFD schemes. In essence, the proposed HEI-SGFD scheme is equivalent to the second-order FD scheme with ordinary grid format. We then determine the HEI-SGFD coefficients in the time–space domain by minimizing the phase velocity error using the least-squares method. Finally, the effectiveness of the proposed method is demonstrated by dispersion analysis and numerical simulations.

Numerical Simulation of a One-Dimentional Non-Linear Wave Equation

In this paper, numerical simulations via regressive and central finite differences of different orders were produced using Fortran code and a one-dimensional non-linear wave equation was solved. The errors obtained during simulations, when using different refinements, were listed and compared in order to determine the validity of the simulation, which demonstrates that the proposed formulation presents satisfactory results.

Time–space domain scalar wave modeling by a novel hybrid staggered-grid finite-difference method with high temporal and spatial accuracies

Staggered-grid finite-difference (SFD) scheme is favored in wave equation simulation due to its superior accuracy and stability to center-grid finite-difference (CFD) scheme. However, for the scalar wave equation (SWE) modeling, the conventional SFD (CSFD) scheme only reaches second-order accuracy in space and time for the discrete SWE; the recently developed modified SFD (MSFD) scheme improves the accuracy to (2N)th-order (N<4) but is costly because the MSFD scheme is coined by adding many extra grid points to the CSFD scheme. To tackle these issues, we develop a cost-effective hybrid SFD (HSFD) scheme, which combines the features of the CSFD and MSFD schemes; we prove that the new HSFD scheme can simultaneously reach (2N)th-order accuracy in space and time for the discrete wave equation. In addition, to deal with the optimization difficulties due to the nonlinear dispersion relation of the SFD schemes, we propose a two-step linear optimization method to improve the accuracy of the new HSFD scheme. The analyses on dispersion, stability properties and numerical simulation examples demonstrate that the new HSFD scheme owns better accuracy and stability than the CSFD scheme. Computational cost analysis shows that the HSFD scheme can be more efficient than the MSFD scheme because it only requires half the additional grid points.

Dual-pairing summation by parts finite difference methods for large scale elastic wave simulations in 3D complex geometries

High-order accurate summation-by-parts (SBP) finite difference (FD) methods constitute efficient numerical methods for simulating large-scale hyperbolic wave propagation problems. Traditional SBP FD operators that approximate first-order spatial derivatives with central-difference stencils often have spurious unresolved numerical wave-modes in their computed solutions. Recently derived high order accurate dual pair SBP operators based non-central (upwind) FD stencils have the potential to suppress these poisonous spurious wave-modes on marginally resolved computational grids. In this paper, we demonstrate that not all high order dual pair SBP FD operators are applicable. Numerical dispersion relation analysis shows that odd-order dual pair SBP FD operators also support spurious unresolved high-frequency wave-modes on marginally resolved meshes. Meanwhile, even-order dual pair SBP FD operators (of order 2,4,6) do not support spurious unresolved high frequency wave-modes and also have better numerical dispersion properties.For all the dual pair SBP FD operators we discretise the three space dimensional (3D) elastic wave equation on boundary-conforming curvilinear meshes. Using the energy method we prove that the semi-discrete approximation is stable and energy-conserving. We derive a priori error estimate and prove the convergence of the numerical error for smooth solutions. Numerical experiments for the 3D elastic wave equation in complex geometries corroborate the theoretical analysis. Numerical simulations of the 3D elastic wave equation in heterogeneous media with complex non-planar free surface topography are given, including numerical simulations of community developed seismological benchmark problems. Computational results show that even-order dual pair SBP FD operators are more efficient, robust and less prone to numerical dispersion errors on marginally resolved meshes when compared to the odd-order dual pair and traditional SBP FD operators. Finally, scaling tests demonstrate nearly perfect strong scaling and verify the efficiency of our parallel implementation of the high order upwind SBP methods.

Adaptive 27-point finite-difference frequency-domain method for wave simulation of 3D acoustic wave equation

The finite-difference frequency-domain (FDFD) method has important applications in the wave simulation of various wave equations. To promote the accuracy and efficiency for wave simulation with the FDFD method, we have developed a new 27-point FDFD stencil for 3D acoustic wave equations. In the developed stencil, the FDFD coefficients depend not only on the ratios of cell sizes in the x-, y-, and z-directions but also depend on the spatial sampling density (SD) in terms of the number of wavelengths per grid. The corresponding FDFD coefficients can be determined efficiently by making use of the plane-wave expression and the lookup table technique. We also develop a new way for designing an adaptive FDFD stencil by directly adding some correction terms to the conventional second-order FDFD stencil, which is simpler to use and easier to generalize. Corresponding dispersion analysis indicates that, compared to the optimal 27-point stencil derived from the average-derivative method (ADM), the developed adaptive 27-point stencil can reduce the required SD from approximately 4 to 2.2 points per wavelength (PPW) for a cubic mesh and to 2.7 PPW for a general cuboid mesh. Numerical examples of a 3D homogeneous model and SEG/EAGE salt-dome model indicate that the developed stencil is more accurate than the ADM 27-point stencil for cubic and general cuboid meshes, while requiring similar CPU time and computational memory as the ADM 27-point stencil for direct and iterative solvers.

Physical properties preserving numerical simulation of stochastic fractional nonlinear wave equation

In this paper, we numerically solve a stochastic space-fractional nonlinear wave equation which includes fractional derivative, nonlinear term, damping term and noise term. The energy of system in the continuous case is derived in detail and discrete energy preserving physical characteristic is also proved. We propose a numerical method that uses Crank-Nicolson difference discretization based on second-order fractional differences. The numerical schemes for stochastic space-fractional nonlinear wave equation with two different noise possess dissipation-preserving energy or energy-conserving under suitable conditions. Moreover, the convergence order of our numerical schemes in time and space is presented in the sense of expectation. Numerical experiments with various types of noise demonstrate that the dissipative property or conservative property coincides with theoretical results.

Effects of channel alternating corrugation on a laser beam propagation in plasmas

The propagation of an intense laser beam in an alternating corrugated plasma channel, which has a wide region and a narrow region in one corrugated space period, is investigated. Compared with the usual corrugated plasma channel, it is found that there are many more resonance peaks and many more abundant beat-like wave phenomena for the laser beam in this extended channel. Moreover, the much narrower region in some special alternating corrugated channels can play the role like a plasma lens for some laser beams, i.e., it can obviously change the laser spot size from small (large) to large (small) amplitude when the laser passes through the narrow region. These results are well confirmed by the final numerical simulations of wave equation and particle-in-cell approach.

Lie group analysis method for the transverse shift of electromagnetic wave on interface of media

The spin Hall effect of electromagnetic wave has important application prospects in the fields of Optical-electromechanical conversion, condensed matter and high energy physics. Based on the Euler–Lagrange equations of electromagnetic field by four-dimensional covariation, a deflection model of electromagnetic wave reflecting and refracting on interface of media is established in this paper. By using Lie group analysis method, the allowable infinitesimal symmetries, conserved quantities of partial differential equations and motion equation of energy center are derived. More importantly, their detailed analytical expressions and properties are given. The content of this paper can reveal the mechanism and influencing factors for transverse shift of the electromagnetic wave energy center, and provide a powerful theoretical basis for the numerical simulation of nonlinear photoelectric wave equation and spin regulation.

Simulation of acoustic wave equation in viscoelastic media by ONADM method

The propagation of acoustic wave in viscoelastic media is of extremely important in seismic exploration, seismology and so on. This paper propose a novel difference scheme for acoustic wave equation in viscoelastic media simulation, named Optimal approximate analytic discretization method (ONADM). This is a new method which can suppress numerical dispersion effectively in larger space step in recent years. The purpose of this paper is to study the propagation of acoustic wave in viscoelastic media by using ONADM method. Finally, The numerical results demonstrate that the ONAD method for acoustic wave equation in viscoelastic media has obvious advantages.