Finite-difference Coefficients Research Articles

Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain

The paper expands the finite difference method to the complex plane, and thus obtains an improvement in the resolution of differential equations with an increase in numerical precision and a generalization in the mathematical modeling of problems. The article begins with a selection of the best techniques for obtaining finite difference coefficients for approximating derivatives in the real domain. Then, the calculation is expanded to the complex domain. The research expands forward, backward, and central difference approximations of the real case by a quadrant approximation in the complex plane, which facilitates the use in boundary conditions of differential equations. The article shows many real and complex finite difference equations with their respective order of error, intended to serve as a basis and reference, which have been tested in practical examples of solving differential equations used in engineering. Finally, a comparison is made between the real and complex techniques of finite difference methods applied in the Theory of Elasticity. As a surprising result, the article shows that the finite difference method has great advantages in numerical precision, diversity of formulas, and modeling generalities in the complex domain when compared to the real domain.

Spatial-temporal high-order rotated-staggered-grid finite-difference scheme of elastic wave equations for TTI medium

In the finite-difference (FD) solution of the elastic wave equation for TTI medium, the traditional FD scheme can improve the computational accuracy of the spatial derivatives. However, the difference discretization on the temporal partial derivatives still uses the FD scheme of second-order accuracy, which cannot guarantee the computational accuracy of the temporal partial derivatives and often causes the temporal dispersion problem. The temporal high-order FD scheme can improve the computational accuracy of the temporal partial derivatives. Nevertheless, due to the complexity of the elastic wave equation in TTI medium, using the conventional staggered-grid technique to solve the elastic wave equation, some spatial derivatives need to be approximated by spatial interpolation, which will become new error and reduce the accuracy of the solution of the equations. Therefore, the currently commonly used temporal high-order FD method for the elastic wave equation in isotropic medium cannot be directly applied to the simulation of anisotropic medium. To solve the above problems, we propose a new temporal high-order rotated-staggered-grid FD stencil and derive a new temporal high-order FD scheme under this new stencil for the TTI medium elastic wave equation. Then we estimated the FD coefficients of the new stencil using Taylor series expansions (TE) and optimized the FD coefficients based on least squares (LS). Numerical experiments of the homogeneous TTI medium model and the modified BP model demonstrate that the TE + LS-based temporal high-order rotated-staggered-grid FD scheme can well solve the temporal dispersion in the numerical simulation of TTI medium.

An approach to remove the numerical dispersion in elastic wave modeling using R-Cycle-GAN networks

Abstract The finite difference forward modeling has been widely used in geophysics exploration and petroleum fields. Because of its high efficiency and easy application for graphical processing units, it has been widely concerned by industry and academia. However, owing to many factors, the problem of numerical dispersion has been an important factor hindering this method. To overcome the numerical dispersion, this paper proposes a method for removing numerical dispersion using deep learning. Unlike the conventional optimized algorithms target to optimize the finite difference coefficients, our strategy is based on big data training to eliminate the dispersion data after seismic data modeling. We design a neural network architecture based on cycle-consistent generative adversarial networks (Cycle-GANs) and residual learning for elastic wave propagation. Under the premise of not significantly increasing the calculation time, we can obtain higher calculation accuracy. Compared with the high-order finite difference algorithm, the calculation time is the advantage of our proposed deep learning method. Tests prove the efficiency and stability of our proposed algorithm.

Neural-Network-Assisted Finite Difference Discretization for Numerical Solution of Partial Differential Equations

A neural-network-assisted numerical method is proposed for the solution of Laplace and Poisson problems. Finite differences are applied to approximate the spatial Laplacian operator on nonuniform grids. For this, a neural network is trained to compute the corresponding coefficients for general quadrilateral meshes. Depending on the position of a given grid point x0 and its neighbors, we face with a nonlinear optimization problem to obtain the finite difference coefficients in x0. This computing step is executed with an artificial neural network. In this way, for any geometric setup of the neighboring grid points, we immediately obtain the corresponding coefficients. The construction of an appropriate training data set is also discussed, which is based on the solution of overdetermined linear systems. The method was experimentally validated on a number of numerical tests. As expected, it delivers a fast and reliable algorithm for solving Poisson problems.

A sixth order quasi-compact finite difference method for Helmholtz equations with variable wave numbers

A sixth order quasi-compact finite difference (SQC) scheme for the Helmholtz equation with variable wave numbers in two and three dimensional rectangular domains has been developed and analyzed. The finite difference coefficients for the solution and the weights for the source term have explicit expressions without involving derivatives of the source term. The proof of the sixth order convergence of the new method is also presented. Numerical experiments presented in this paper confirmed the order of convergence of the new method.

Reducing the low-wavenumber dispersion error by building the Lagrange dual problem with a powerful local restriction

Abstract The broadband finite-difference (FD) coefficients computed by a cost function have been widely applied to suppress of numerical dispersion. Under the same conditions, the FD coefficients with small low-wavenumber dispersion error will produce a more accurate numerical solution in the long-time seismic wave simulation. Thus, how to reduce the low-wavenumber dispersion error becomes a crucial problem. According to the research into the zero-point position at the dispersion curve for three types of cost function, we found that the more zero points concentrate on the low-wavenumber region, the less the dispersion error. Therefore, the concentration of zero points is a good way to reduce dispersion error, which can be implemented by modified wavenumbers of zero points. Then, we design a Lagrange dual problem with a restriction based on the modified wavenumbers. The requirements for constructing the Lagrange dual problem are the optimization function and restricted condition, where the former is based on the dispersion relation, and the latter comprises the modified wavenumbers. The solution of this optimization problem, calculated by the dual ascent method, affords a less low-wavenumber dispersion error than the solution yielded by the alternating direction method of multipliers (ADMM). The theoretical analysis and numerical modeling suggest that the proposed method is superior to the existing optimal FD coefficients in reducing numerical error accumulation in low-frequency simulation.

Efficient temporal high-order staggered-grid scheme with a dispersion-relation-preserving method for the scalar wave modeling

Staggered-grid finite-difference (FD) method is widely used to solve the wave equation for the numerical seismic modeling, and it is a key step of the advanced seismic imaging and inversion problem. However, the conventional FD method is prone to instability and dispersion error due to the insufficient approximation accuracy. In this work, we propose an efficient temporal high-order finite-difference (FD) scheme with the cross-rhombus stencil. Compared with the standard cross-rhombus method, the new method has less computational cost due to we simplify the FD scheme. Moreover, the dispersion relation of the new method is easy to obtain the dispersion-relation-preserving (DRP) FD coefficients, which can significantly alleviate the spatial and temporal dispersion errors. Dispersion and stability analyses indicate that the new scheme has better performance in seismic modeling than the conventional method, and numerical experiments also indicate that the new scheme can effectively mitigate dispersion error and improve the numerical accuracy.

Controlled-source electromagnetic modeling using a high-order finite-difference time-domain method on a nonuniform grid

Simulation of 3D low-frequency electromagnetic (EM) fields propagating in the earth is computationally expensive. We have developed a fictitious wave-domain high-order finite-difference time-domain (FDTD) modeling method on nonuniform grids to compute frequency-domain 3D controlled-source EM data. The method overcomes the inconsistency issue widely present in the conventional second-order staggered-grid finite-difference scheme over a nonuniform grid, achieving high accuracy with an arbitrarily high-order scheme. The finite-difference coefficients adaptive to the node spacings can be accurately computed by inverting a Vandermonde matrix system using an efficient algorithm. A generic stability condition applicable to nonuniform grids is established, revealing the dependence of the time step and these finite-difference coefficients. A recursion scheme using fixed-point iterations is designed to determine the stretching factor to generate the optimal nonuniform grid. The grid stretching in our method reduces the number of grid points required in the discretization, making it more efficient than the standard high-order FDTD with a densely sampled uniform grid. Instead of stretching in vertical and horizontal directions, better accuracy of our method is observed when the grid is stretched along the depth without horizontal stretching. The efficiency and accuracy of our method are demonstrated by numerical examples.

Wave Equation Numerical Simulation and RTM With Mixed Staggered-Grid Finite-Difference Schemes

For the conventional staggered-grid finite-difference scheme (C-SFD), although the spatial finite-difference (FD) operator can reach 2Mth-order accuracy, the FD discrete wave equation is the only second-order accuracy, leading to low modeling accuracy and poor stability. We proposed a new mixed staggered-grid finite-difference scheme (M-SFD) by constructing the spatial FD operator using axial and off-axial grid points jointly to approximate the first-order spatial partial derivative. This scheme is suitable for modeling the stress–velocity acoustic and elastic wave equation. Then, based on the time–space domain dispersion relation and the Taylor series expansion, we derived the analytical expression of the FD coefficients. Theoretically, the FD discrete acoustic wave equation and P- or S-wave in the FD discrete elastic wave equation given by M-SFD can reach the arbitrary even-order accuracy. For acoustic wave modeling, with almost identical computational costs, M-SFD can achieve higher modeling accuracy than C-SFD. Moreover, with a larger time step used in M-SFD than that used in C-SFD, M-SFD can achieve higher computational efficiency and reach higher modeling accuracy. For elastic wave simulation, compared to C-SFD, M-SFD can obtain higher modeling accuracy with almost the same computational efficiency when the FD coefficients are calculated based on the S-wave time–space domain dispersion relation. Solving the split elastic wave equation with M-SFD can further improve the modeling accuracy but will decrease the efficiency and increase the memory usage as well. Stability analysis shows that M-SFD has better stability than C-SFD for both acoustic and elastic wave simulations. Applying M-SFD to reverse time migration (RTM), the imaging artifacts caused by the numerical dispersion are effectively eliminated, which improves the imaging accuracy and resolution of deep formation.

Optimizing finite-difference scheme in multidirections on rectangular grids based on the minimum norm

The finite-difference (FD) method is widely used in numerical simulation; however, its accuracy suffers from numerical spatial dispersion and numerical anisotropy. The single-direction optimization methods, which optimize the FD coefficients along a single spatial direction, can suppress numerical spatial dispersion, but they are suboptimal for mitigating numerical anisotropy on rectangular grids. We have developed a multidirection optimization method that penalizes approximation errors among all propagation angles on rectangular grids with the minimum norm (i.e., [Formula: see text] norm) to mitigate numerical spatial dispersion and numerical anisotropy simultaneously. Given maximum absolute error tolerance and grid-spacing ratio, we first determine the optimal order of the FD operator in each spatial direction. Then, we penalize approximation errors within the wavenumber-azimuth domain to obtain the optimized FD coefficients. Theoretical analysis and numerical experiments find that our method is superior to single-direction optimization methods in suppressing numerical spatial dispersion and mitigating numerical anisotropy for square and rectangular grids. For homogeneous models with grid-spacing ratios of 1.0 (i.e., square grids), 1.2, and 1.4, the root-mean-square (rms) errors obtained by our method are 77%, 80%, and 72% that of the single-direction optimization method adopting the [Formula: see text] norm, respectively. For the Marmousi model with a grid-spacing ratio of 1.4, the rms error of our method is 36% that of the single-direction optimization method based on the [Formula: see text] norm. Such an evident improvement on error suppression is critical for numerical simulations adopting more flexible grid-spacing ratios.

An optimized finite-difference method to minimize numerical dispersion of acoustic wave propagation using a genetic algorithm

The finite-difference method (FDM) is one of the most popular methods for numerical simulation of wave propagation. The major challenge that we face in solving a partial differential equation for the wave propagation is numerical dispersion. We use an automated and optimized FDM using a genetic algorithm (GA) to optimally compute second-order spatial derivatives. In our method, the explicit finite-difference coefficients are calculated using the GA to minimize the dispersion (phase velocity) for all wavenumbers without using any specific window function. The amplitudes of the pseudospectral window (finite difference [FD] coefficients) are optimized by making the phase velocity close to the analytical solution at each wavenumber and stability close to that of the conventional FDM. Although FD coefficients in this method depend on velocity, grid spacing, and time step, less dispersive solutions can be achieved by computing suitable FD coefficients for varying cases. It takes only a few seconds of additional computation time depending on the order of approximation (e.g., approximately 20 s for the 12th order of approximation in a machine with 120 GB RAM and 2.0 GHz processor). Comparison of the numerical results from our algorithm with those from an existing pseudospectral method, the conventional FDM, and an existing time-space optimized FDM indicates that the proposed method is less dispersive for any order of approximation.

Higher order differences on arbitrary discrete time scales and related generating functions

In this paper we uncover some fundamental relationships involving the weights one needs to calculate the n-th difference of a function on an arbitrary discrete time scale. One of several interesting results we obtain is a formula that allows one to calculate the value of any desired finite difference coefficient directly from the graininess function for the time scale under consideration. This foundational work beautifully combines analysis, algebra, and combinatorics to obtain this and other interesting results. Some of the other interesting results include (i) for a fixed n, the n-th finite difference coefficients sum to 0 and (ii) there are nice and useful generating functions that encode various sequences of finite difference coefficients. Throughout we show the results coincide with well-known results in the special cases where the time scales are either quantum time scales or time scales with constant graininess.

Removing the stability limit of the time-space domain explicit finite-difference schemes for acoustic modeling with stability condition-based spatial operators

The time step and grid spacing in explicit finite-difference (FD) modeling are constrained by the Courant-Friedrichs-Lewy (CFL) condition. Recently, it has been found that the spatial FD coefficients may be designed through simultaneously minimizing the spatial dispersion error and maximizing the CFL number. This allows one to stably use a larger time step or a smaller grid spacing than usually possible. However, when using such a method, only second-order temporal accuracy is achieved. To address this issue, I propose a new method to determine the spatial FD coefficients, which simultaneously satisfy the stability condition of the whole wavenumber range and the time-space domain dispersion relation of a given wavenumber range. Therefore, stable modeling can be performed with high-order spatial and temporal accuracy. The coefficients can adapt to the variation of velocity in heterogeneous models. In addition, based on the hybrid absorbing boundary condition, I develop a strategy to stably and effectively suppress artificial reflections from the model boundaries for large CFL numbers. Stability analysis, accuracy analysis, and numerical modeling demonstrate the accuracy and effectiveness of my proposed method.

A novel equivalent staggered-grid finite-difference scheme and its optimization strategy for variable-density acoustic wave modelling

Compared with the standard staggered-grid finite-difference (FD) methods, equivalent staggered-grid (ESG) ones can significantly reduce the computational memory for acoustic wave modelling in the variable-density media. To further enhance the simulation efficiency and accuracy, one way is to optimize the FD coefficients, another way is to design new FD stencils. In this paper, we propose a modified ESG (M-ESG) scheme which can significantly accelerate the wavefield simulation process while preserving or even improving the modelling accuracy. We calculate the FD coefficients by approximating the temporal and spatial derivatives simultaneously based on time–space domain (TS-D) dispersion relation of the discrete wave equation. Our M-ESG scheme in the TS-D can maintain basically the same accuracy as the conventional ESG (C-ESG) one when the FD coefficients are derived by the Taylor-series expansion (TE) approach. Note that the TS-D dispersion relation is nonlinear with respect to the FD coefficients of the C-ESG scheme, so it is difficult to obtain the optimized FD coefficients for the discrete wave equation. However, we can minimize the L2-norm error of the dispersion relation based on our M-ESG scheme to implement a linear FD coefficients optimization strategy, which is easy and efficient. Comparisons with TE- and optimization-based C-ESG schemes demonstrate the accuracy, stability, and efficiency superiorities of our TE- and optimization-based M-ESG ones.

A high-order finite-difference scheme for time-domain modeling of time-varying seismoelectric waves

Simulation of the seismoelectric effect serves as a useful tool to capture the observed seismoelectric conversion phenomenon in porous media, thus offering promising potential in underground exploration activities to detect pore fluids such as water, oil, and gas. The static electromagnetic (EM) approximation is among the most widely used methods for numerical simulation of the seismoelectric responses. However, the static approximation ignores the accompanying electric field generated by the shear wave, resulting in considerable errors when compared to analytical results, particularly under high-salinity conditions. To mitigate this problem, we have adopted a spatial high-order finite-difference time-domain method based on Maxwell’s full equations of time-varying EM fields to simulate the seismoelectric response in 2D mode. To improve the computational efficiency influenced by the velocity differences between seismic and EM waves, different time steps are set according to the stability conditions, and the seismic feedback values of EM time nodes are obtained by linear approximation within the seismic unit time step. To improve the simulation accuracy of the seismoelectric response with the time-varying EM calculation method, finite-difference coefficients are obtained by solving the spatial high-order difference approximation based on the Taylor expansion. Our method yields consistent simulation results compared to those obtained from the analytical method under different salinity conditions, thus indicating its validity for simulating seismoelectric responses in porous media. We further apply our method to layered and anomalous body models and extend our algorithm to three dimensions. Results indicate that the time-varying EM calculation method can effectively capture the reflection and transmission phenomena of the seismic and EM wavefields at the interfaces of contrasting media. This may allow for the identification of abnormal locations, thus highlighting the capability of seismoelectric response simulation to detect subsurface properties.

Analysis of New RBF‐FD Weights, Calculated Based on Inverse Quadratic Functions

Local radial basis functions (RBFs) have many advantages for solution of differential equations. In some of these radial functions, there is a parameter that has a special effect on the accuracy of the answer and is known as the shape parameter. In this article, first of all, we derive inverse quadratic (IQ)‐based RBF‐generated finite difference coefficients for some derivatives in one dimension (1D). Then, to evaluate the efficiency of these new weights and also the effect of the shape parameter on the accuracy of the resulting approximations, we will test them with a suitable function. After that, we focus on solving some boundary value problems (BVPs), using IQ‐based RBF‐FD method. There is a range for the shape parameter in which the approximation error is less than other areas. We use an efficient algorithm to find the best value of the RBF parameter for the problem domain. Our studies show that IQ‐based RBF‐FD weights could be derived analytically easier than multiquadrics (MQs) which were previously presented in the literature. Besides, the results of numerical examples confirm the high accuracy of these new formulas. For better comparison, we revisit some previously studied illustrative examples.

Acoustic wave propagation with new spatial implicit and temporal high-order staggered-grid finite-difference schemes

AbstractThe implicit staggered-grid (SG) finite-difference (FD) method can obtain significant improvement in spatial accuracy for performing numerical simulations of wave equations. Normally, the second-order central grid FD formulas are used to approximate the temporal derivatives, and a relatively fine time step has to be used to reduce the temporal dispersion. To obtain high accuracy both in space and time, we propose a new spatial implicit and temporal high-order SG FD stencil in the time–space domain by incorporating some additional grid points to the conventional implicit FD one. Instead of attaining the implicit FD coefficients by approximating spatial derivatives only, we calculate the coefficients by approximating the temporal and spatial derivatives simultaneously through matching the dispersion formula of the seismic wave equation and compute the FD coefficients of our new stencil by two schemes. The first one is adopting a variable substitution-based Taylor-series expansion (TE) to derive the FD coefficients, which can attain (2M + 2)th-order spatial accuracy and (2N)th-order temporal accuracy. Note that the dispersion formula of our new stencil is non-linear with respect to the axial and off-axial FD coefficients, it is complicated to obtain the optimal spatial and temporal FD coefficients simultaneously. To tackle the issue, we further develop a linear optimisation strategy by minimising the L2-norm errors of the dispersion formula to further improve the accuracy. Dispersion analysis, stability analysis and modelling examples demonstrate the accuracy, stability and efficiency advantages of our two new schemes.

Applying an advanced temporal and spatial high-order finite-difference stencil to 3D seismic wave modeling

Present finite-difference (FD) algorithms for modeling seismic wave propagation in 3D acoustic media are mainly based on the traditional orthogonal stencil and can achieve spatial high-order but temporal second-order accuracy. Therefore, these approaches may result in visible temporal dispersion and even instability for relatively large time sampling intervals. In this paper, we develop an advanced FD stencil with temporal and spatial arbitrary even-order accuracy to solve the 3D acoustic wave equation. The temporal derivative is approximated by an octahedron stencil with the operator length N together with the traditional second-order FD. Meanwhile, the spatial derivatives are calculated by the orthogonal stencil with the operator length M. The plane-wave theory is introduced to derive the time-space-domain dispersion relation and estimate the corresponding high-order FD coefficients by Taylor expansion. To achieve higher accuracy, we further optimize FD coefficients by applying least-squares (LS) to minimize the relative error of the time-space-domain dispersion relation. Dispersion and stability analyses show that our new schemes have higher modeling accuracy and propagation stability than the conventional method. Numerical examples demonstrate that the proposed FDs can generate greater temporal accuracy on the prerequisite of preserving satisfactory spatial accuracy. Our new temporal high-order FD stencil by incorporating larger time step, shorter operator lengths, LS-based coefficients and variable-length operators can be regarded as a kind of accurate and efficient tool for seismic wave modeling.

Viscoelastic Wave Simulation with High Temporal Accuracy Using Frequency-Dependent Complex Velocity

In recent decades, the study of seismic attenuation has received more and more concerns because it can stimulate the development of wave propagation simulation and improve the accuracy of structure imaging and reservoir prediction. In this paper, we review the attenuation theory and the development of high temporal accuracy wave simulation. The conventional mathematical models to describe the characteristics of viscoelastic are based on constant-Q model or standard linear solids theory. However, these approaches possess some noticeable shortcomings. Therefore, we introduce a frequency-dependent complex velocity to derive the novel viscoelastic wave equations with decoupled amplitude dissipation and phase dispersion. To obtain high temporal accuracy viscoelastic wave simulation, we adopt the normalized pseudo-Laplacian to compensate for the temporal dispersion errors caused by the second-order finite-difference discretization in the time domain. During the implementation, we incorporate the normalized pseudo-Laplacian into the optimized staggered-grid finite-difference coefficients. Therefore, it can greatly reduce the times of low-rank decomposition and Fourier transform and largely improve the computational efficiency. Based on this strategy, we can implement the high temporal accuracy viscoelastic wavefield extrapolation by comprehensively exploiting the staggered-grid finite-difference scheme, pseudo-spectral method and low-rank decomposition algorithm. Meanwhile, a linear velocity model is employed to evaluate the accuracy of low-rank approximation. Furthermore, we use several numerical examples to carry out the comparison between our scheme and other conventional methods. The numerical results reveal that our proposed scheme can effectively compensate for temporal dispersion errors and help generate high temporal accuracy viscoelastic wave solutions.

A consistent implementation of point sources on finite-difference grids

SUMMARY We present a method to position point sources at arbitrary locations on finite-difference (FD) grids. We show that implementing point sources on single nodes can cause considerable errors when modelling with the FD method. In contrast, we propose to create a spatially distributed source (over multiple nodes) that nonetheless creates the desired point-source response. The spatial point source is formulated in the wavenumber domain and is based on the FD coefficients used for the wave propagation. Using this ‘FD-consistent source’ on 1-D and 2-D examples, we show that we can obtain superior fits to analytical solutions compared to single-node or sinc-function source implementations, and we show that sources can be offset to arbitrary locations from ‘on’ the grid to ‘off’ the grid, while resulting in solutions that are identical to within machine precision