Lax-Wendroff Correction Research Articles

A Strong Stability-Preserving Predictor-Corrector Method for the Simulation of Elastic Wave Propagation in Anisotropic Media

AbstractIn this paper, we propose a strong stability-preserving predictor-corrector (SSPC) method based on an implicit Runge-Kutta method to solve the acoustic- and elastic-wave equations. We first transform the wave equations into a system of ordinary differential equations (ODEs) and apply the local extrapolation method to discretize the spatial high-order derivatives, resulting in a system of semi-discrete ODEs. Then we use the SSPC method based on an implicit Runge-Kutta method to solve the semi-discrete ODEs and introduce a weighting parameter into the SSPC method. On top of such a structure, we develop a robust numerical algorithm to effectively suppress the numerical dispersion, which is usually caused by the discretization of wave equations when coarse grids are used or geological models have large velocity contrasts between adjacent layers. Meanwhile, we investigate the performance of the SSPC method including numerical errors and convergence rate, numerical dispersion, and stability criteria with different choices of the weighting parameter to solve 1-D and 2-D acoustic- and elastic-wave equations. When the SSPC is applied to seismic simulations, the computational efficiency is also investigated by comparing the SSPC, the fourth-order Lax-Wendroff correction (LWC) method, and the staggered-grid (SG) finite difference method. Comparisons of synthetic waveforms computed by the SSPC and analytic solutions for acoustic and elastic models are given to illustrate the accuracy and the validity of the SSPC method. Furthermore, several numerical experiments are conducted for the geological models including a 2-D homogeneous transversely isotropic (TI) medium, a two-layer elastic model, and the 2-D SEG/EAGE salt model. The results show that the SSPC can be used as a practical tool for large-scale seismic simulation because of its effectiveness in suppressing numerical dispersion even in the situations such as coarse grids, strong interfaces, or high frequencies.

High accuracy wave simulation – Revised derivation, numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation

The nearly analytic integration discrete (NAID) method for solving the two-dimensional acoustic wave equation has been fully mathematically revised, analyzed and tested. The NAID method is an alternative numerical modeling method for generating synthetic seismograms. The acoustic wave equation is first transformed into a system of first-order ordinary differential equations (ODEs) with respect to time variable t, and then directly integrated at a small time interval of [tn,tn+1] to obtain semi-discrete ordinary differential equations. The integral kernel is expanded into a truncated Taylor series, to which the integration operator is explicitly applied. The high-order temporal derivatives involved in the integral kernel are replaced by high-order spatial derivatives, which then are approximately calculated as a weighted linear combination of the displacement, the particle-velocity, and their spatial gradients. In this article, we investigate the theoretical properties of the revised NAID method, including the discrete error and the stability criteria. Numerical results for constant and layered velocity models show that, comparing to the Lax–Wendroff correction (LWC) scheme and the staggered-grid finite difference method, the NAID method can effectively suppress the numerical dispersion and source-noises caused by the discretization of the acoustic wave equation with too-coarse spatial grids or when models have strong velocity contrasts between adjacent layers. The proposed NAID method has been applied in computing the acoustic wavefields for two heterogeneous models – the corner edge model and the Marmousi model. Promising numerical results illustrate that the NAID method provides an encouraging tool for large-scale and complex wave simulation and inversion problems based on the acoustic equation.

A Split-Step Algorithm for Effectively Suppressing the Numerical Dispersion for 3D Seismic Propagation Modeling

In this article, we propose a 3D split-step algorithm (SSA) for seismic- wave simulation. We first transform the wave equations in 3D anisotropic media into a system of first-order partial differential equations with respect to time t. Then we use the multidimensional high-order interpolation method to approximate the high-order spatial derivatives, so that we obtain a system of semidiscrete ordinary differential equations (ODEs). Finally, the third-order implicit Adams method and truncated dif- ferentiator series method are applied to solve the semidiscrete ODEs. We provide the theoretical study on the properties of the 3D SSA, such as stability criteria, theoretical error, numerical error, numerical dispersion, and computational efficiency. We also compare some seismic modeling results of this method against those of some high-order finite-difference schemes. Theoretical analysis and numerical tests show that the 3D SSA is third-order accurate in time and fourth-order accurate in space. However, its computational costs and memory requirements are much less than those of the fourth-order Lax-Wendroff correction method and the fourth-order staggered- grid method. Using a multilayer elastic model with large velocity contrasts and free surface, we compare the result of the 3D SSA with that of the discrete-wavenumber method. We also present the synthetic seismograms in the 3D three-layer isotropic medium, the wave-field snapshots in the 3D two-layer medium, and the 3D trans- versely isotropic medium with a vertical symmetry axis. All these promising numer- ical results illustrate that the 3D SSA can suppress effectively the numerical dispersion caused by discretizing the wave equations when too few sampling points per mini- mum wavelength are used or when models have large velocity contrasts between adjacent layers, further resulting in both increasing the computational efficiency and saving the storage space when big grids are used.

Simulation of acoustic wavefields in heterogeneous media: A robust method for automatic suppression of numerical dispersion

Numerical dispersion limits the application of numerical simulation methods for solving the acoustic wave equation in large-scale computation. The nearly analytic discrete method (NADM) and its improved version for suppressing numerical dispersion were developed recently. This new method is a refinement of two previous methods and further increases the ability of suppressing numerical dispersion for modeling acoustic wave propagation in 2D heterogeneous media, which uses acoustic wave displacement, particle velocity, and their gradients to restructure the acoustic wavefield via the truncated Taylor expansion and the high-order interpolation approximate method. For the method proposed, we investigate its implementation and compare it with the higher-order Lax-Wendroff correction (LWC) scheme, the original nearly analytic discrete method (NADM) and its im-proved version with regard to numerical dispersion, computational costs, and computer storage requirements. The numerical dispersion relations provided by the refined algorithm for 1D and 2D cases are analyzed, as well as the numerical results obtained by this method against the exact solution for the 2D acoustic case. Numerical results show that the refined method gives no visible numerical dispersion for very large spatial grid increments. It can simulate high-frequency wave propagation for a given grid interval and automatically suppress the numerical dispersion when the acoustic wave equation is discretized, when too few samples per wavelength are used, or when models have large velocity contrasts. Unlike the high-order LWC methods, our present method can save substantial computational costs and memory requirements because very large grid increments can be used. The refined method can be used for the simulation of large-scale wavefields.

An Explicit Method Based on the Implicit Runge-Kutta Algorithm for Solving Wave Equations

Abstract A new explicit differentiator series method based on the implicit Runge–Kutta method, called the IRK-DSM in brief, is developed for solving wave equations. To develop the new algorithm, we first transform the wave equation, usually described as a partial differential equation (PDE), into a system of first-order ordinary differential equations (ODEs) with respect to time t . Then we use a truncated differentiator series method of the implicit Runge–Kutta method to solve the semidiscrete ordinary differential equations, while the high-order spatial derivatives included in the ODEs are approximated by the local interpolation method. We analyze the theoretical properties of the IRK-DSM, including the stability criteria for solving the 1D and 2D acoustic-wave equations, numerical dispersion, discretizing error, and computational efficiency when using the IRK-DSM to model acoustic-wave fields. For comparison, we also present the stability criteria and numerical dispersion of the so-called Lax–Wendroff correction (LWC) methods with the fourth-order and eighth-order accuracies for the 1D case. Promising numerical results show that the IRK-DSM provides a useful tool for large-scale practical problems because it can effectively suppress numerical dispersions and source-noises caused by discretizing the acoustic- and elastic-wave equations when too-coarse grids are used or the models have a large velocity contrast between adjacent layers. Theoretical analysis and numerical modeling also demonstrate that the IRK-DSM, through combining both the implicit Runge–Kutta scheme with good stability condition and the approximate differentiator series method, is a robust wave-field modeling method.

An explicit split-step algorithm of the implicit Adams method for solving 2-D acoustic and elastic wave equations

SUMMARY We first transform the seismic wave equations in 2-D inhomogeneous anisotropic media into a system of first-order partial differential equations with respect to time t. And then we develop a new explicit split-step algorithm (SSA) to solve the transformed equations through using the implicit Adams method and high-order interpolation approximation. Our algorithm enables wave propagation to be simulated in two dimensions through generally isotropic and anisotropic models. The high-order space derivatives in this SSA are determined by using the wave displacement, particle-velocity and its gradients simultaneously, while the time advancing of modelling seismic propagation uses the slit-operator algorithm based on the third-order implicit Adams method. On the basis of such a structure, the so-called SSA can suppress effectively the numerical dispersion and source-generated noises caused by discretizing the wave equations when too-coarse grids are used, and is third-order accuracy in time and fourth-order accuracy in space. We explore the theoretical properties of the SSA including the stability criteria of the SSA for solving 1-D and 2-D scalar wave equations, numerical dispersion, theoretical error and numerical error and computational efficiency when using the SSA to model the acoustic wave fields. Meanwhile, we also present the synthetic seismograms generated by the SSA in the three-layer isotropic medium and the 2-D elastic two-layer isotropic and anisotropic wavefield snapshots computed by the fourth-order Lax–Wendroff correction (LWC) scheme and the SSA, respectively. Numerical calculations of the relative errors show that the numerical error of the SSA is less than those of the conventional finite difference (FD) method and the fourth-order LWC scheme. Promising numerical results further illustrate that the SSA has less numerical dispersions and can suppress effectively the source-generated noises, further resulting in both increasing the computational efficiency and saving the space storage.

Elastic wave modelling method based on the displacement–velocity fields: an improving nearly analytic discrete approximation

In this paper, we present an improvement to our previously published nearly analytic discrete method (NADM) to solve acoustic and elastic wave equations. We compare the numerical errors of the improved NADM with the original NADM and also the fourth-order Lax–Wendroff correction and present examples of three-component wave fields in 2D transversely isotropic media with strong velocity contrasts. Comparing with the original NADM, we find that the improved method requires significantly less storage space and can increase the time accuracy from second order of the original NADM to fourth order, while the space accuracy remains the same as that of the original one. Theoretical analyses and numerical results suggest that our improved NADM is suitable for large-scale numerical modelling as it can effectively suppress numerical dispersion and source-generated noises caused by discretising wave equations when too-coarse grids are used.

Optimally Accurate Nearly Analytic Discrete Scheme for Wave-Field Simulation in 3D Anisotropic Media

We present a new numerical method for elastic wave modeling in 3D isotropic and anisotropic media, which is called the 3D optimal nearly analytic dis- crete method (ONADM) in this article. This work is an extension of the 2D ONADM (Yang et al., 2004) that models acoustic and elastic waves propagating in 2D media. The formulation is derived by using a multivariable truncated Taylor series expansion and high-order interpolation approximations. Our 3D ONADM enables wave propa- gation to be simulated in three dimensions through isotropic and anisotropic models. Promising numerical results show that the error of the ONADM for the 3D case is less than those of the conventional finite-difference (FD) method and the so-called Lax-Wendroff correction (LWC) schemes, measured quantitatively by the root- mean-square deviation from analytical solution. The seismic wave fields in the 3D isotropic and anisotropic media are simulated and compared with those obtained by using the fourth-order LWC method and exact solutions for the acoustic wave case. We show that, compared with the conventional FD method and the LWC schemes, the ONADM for the 3D case can reduce significantly the storage space and compu- tational costs. Numerical experiments illustrate that the ONADM provides a useful tool for the 3D large-scale isotropic and anisotropic problems and it can suppress effectively numerical dispersions caused by discretizing the 3D wave equations when too-coarse grids are used, which is the same as the 2D ONADM. Numerical modeling also implies that simultaneously using both the wave displacement and its gradients to approximate the high-order derivatives is important for both decreasing the nu- merical dispersion caused by the discretization of wave equations and compensating the important wave-field information included in wave-displacement gradients.

An improved nearly analytical discrete method: an efficient tool to simulate the seismic response of 2-D porous structures

The nearly analytic discrete method (NADM) for acoustic and elastic waves in porous elastic media is a perturbation method recently proposed by Yang et al (2006a Commun. Comput. Phys. 1 528–47). This method uses the truncated Taylor series expansion to approximate the time derivatives and the local high-order interpolation to approximate the spatial high-order derivatives by simultaneously using the displacements and its gradients and the velocity. As a result, it can suppress effectively numerical dispersions caused by the discretizing the wave equations when too-coarse grids are used. In this paper, we present an improved nearly-analytic discrete method (INADM) for the porous case. We compare numerically the error of the INADM with those of the original NADM and the so-called Lax–Wendroff correction (LWC) schemes for 1-D and 2-D cases, and give the wave-field modelling in 2-D porous isotropic and anisotropic media. We show that, compared with the original NADM, the INADM for the 2-D case can reduce significantly the storage space and increase time accuracy, while the space accuracy remains the same as that of the original one. Numerical experiments show that the error of the INADM for the porous case is less than those of the NADM and the fourth-order LWC scheme. The three-component seismic wave-fields in the 2-D porous isotropic medium are compared with those obtained by using the NADM, the LWC method, and exact solutions. Several characteristics of waves propagating in porous anisotropic media, computed by the INADM, are also reported in this study. Promising numerical results illustrate that the INADM provides a useful tool for large-scale porous problems and it can effectively suppress numerical dispersions.

A modified Lax-Wendroff correction for wave propagation in media described by Zener elements

SUMMARY A modified Lax-Wendroff correction for wave propagation in attenuating and dispersive media described by Zener elements is presented. As opposed to the full correction, this new technique is explicit and offers large computational savings. The technique may be applied to a wide variety of hyperbolic problems. Here, the concept is illustrated for wave propagation in visco-acoustic media.