Optimal implicit staggered‐grid finite‐difference schemes based on the sampling approximation method for seismic modelling

Optimal rotated staggered-grid finite-difference schemes for elastic wave modeling in TTI media

As a differential method, Finite-Difference TimeDomain (FDTD) has its inherent numerical dispersion error characteristics, which raises concerns about its reliability in handling phase issues. To address the dispersion performance, scholars have proposed an improved FDTD algorithm based on a hexagonal grid FDTD (H-FDTD). In two-dimensional scenarios, this algorithm discretizes continuous space using a hexagonal grid and constructs the FDTD difference scheme based on directional derivatives, thereby formulating the dispersion equation and iterative equations. Although the numerical dispersion relationship is analyzed using Taylor series expansion, and carefully selected coefficients significantly optimize the numerical anisotropy of the H-FDTD algorithm, there remains a certain deviation between the magnitude of the numerical phase velocity and the analytical solution. To this end, this paper proposes a low numerical dispersion H-FDTD method. By introducing correction coefficients to adjust the permittivity and permeability in the H-FDTD algorithm, it is possible to maintain the isotropic of numerical dispersion while further reducing numerical dispersion errors, thereby achieving precise phase velocity at a single frequency point.

The electrostatic potential which arises from a lattice array of point ions is computed in terms of a Taylor's series expansion for small distances from a lattice site. This expansion gives the change in electrostatic energy when an ion moves in the background of a perfect point ion lattice potential. The Taylor's series coefficients for terms up to fourth order in the ion displacement are evaluated for the NaCl and CaF2 lattice structures.

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

Numerical simulation of three-dimensional (3D) seismic wavefields forms the basis of the research on the migration methods of 3D seismic data based on wave equations. Because the simulation precision of wavefield extrapolation determines the imaging accuracy to a certain extent, it is very important to study how to enhance the forward modeling precision of 3D seismic wavefields. Thus, we build on an optimized 3D staggered-grid finite-difference (SFD) method with high simulation precision based on two-dimensional (2D) seismic modeling. Since it generates the corresponding difference coefficients by utilizing the least square (LS) method to minimize the objective function constructed by the time-space domain dispersion relation of the 3D acoustic wave equation, our optimized time-space domain LS-based 3D SFD method can effectively enhance the modeling precision of the 3D seismic wavefields in theory compared with the 3D SFD methods based on the Taylor-series expansion (TE), especially for the large wavenumber range. Examining the numerical dispersion, algorithm stability and computational cost, we compare our optimized time-space domain LS-based 3D SFD method with three conventional TE-based and LS-based 3D SFD methods to illustrate and demonstrate its effectiveness and feasibility. The numerical examples from different 3D models suggest that our optimized time-space domain LS-based 3D SFD method can generate less numerical dispersion and higher simulation accuracy for 3D seismic wavefields than three other conventional 3D SFD methods, but its stability condition is stricter and its computational cost is slightly higher.

This study applies the space–time generalized finite difference scheme to solve nonlinear dispersive shallow water waves described by the modified Camassa–Holm equation, the modified Degasperis–Procesi equation, the Fornberg–Whitham equation, and its modified form. The proposed meshless numerical scheme combines the space–time generalized finite difference method, the two-step Newton’s method, and the time-marching method. The space–time approach treats the temporal derivative as a spatial derivative. This enables the discretization of all partial derivatives using a spatial discretization method and efficiently handles mixed derivatives with the proposed mesh-less numerical scheme. The space–time generalized finite difference method is derived from Taylor series expansion and the moving least-squares method. The numerical discretization process only involves functional data and weighting coefficients on the central and neighboring nodes. This results in a sparse matrix system of nonlinear algebraic equations that can be efficiently solved using the two-step Newton’s method. Additionally, the time-marching method is employed to advance the space–time domain along the time axis. Several numerical examples are presented to validate the effectiveness of the proposed space–time generalized finite difference scheme.

Interpolation Schemes for Three-Dimensional Velocity Fields from Scattered Data Using Taylor Expansions

Staggered-grid finite-difference (SFD) methods are widely used in modeling seismic-wave propagation, and the coefficients of finite-difference (FD) operators can be estimated by minimizing dispersion errors using Taylor-series expansion (TE) or optimization. We developed novel optimal time-space-domain SFD schemes for acoustic- and elastic-wave-equation modeling. In our schemes, a fourth-order multiextreme value objective function with respect to FD coefficients was involved. To yield the globally optimal solution with low computational cost, we first used variable substitution to turn our optimization problem into a quadratic convex one and then used least-squares (LS) to derive the optimal SFD coefficients by minimizing the relative error of time-space-domain dispersion relations over a given frequency range. To ensure the robustness of our schemes, a constraint condition was imposed that the dispersion error at each frequency point did not exceed a given threshold. Moreover, the hybrid absorbing boundary condition was applied to remove artificial boundary reflections. We compared our optimal SFD with the conventional, TE-based time-space-domain, and LS-based SFD schemes. Dispersion analysis and numerical simulation results suggested that the new SFD schemes had a smaller numerical dispersion than the other three schemes when the same operator lengths were adopted. In addition, our LS-based time-space-domain SFD can obtain the same modeling accuracy with shorter spatial operator lengths. We also derived the stability condition of our schemes. The experiment results revealed that our new LS-based SFD schemes needed a slightly stricter stability condition.

Identification of Beam Boundary Conditions Using Displacement Derivatives Estimations

We develop an alternative formulation of conservative finite difference weighted essentially non-oscillatory (WENO) schemes to solve conservation laws. In this formulation, the WENO interpolation of the solution and its derivatives are used to directly construct the numerical flux, instead of the usual practice of reconstructing the flux functions. Even though this formulation is more expensive than the standard formulation, it does have several advantages. The first advantage is that arbitrary monotone fluxes can be used in this framework, while the traditional practice of reconstructing flux functions can be applied only to smooth flux splitting. The second advantage, which is fully explored in this paper, is that it is more straightforward to construct a Lax-Wendroff time discretization procedure, with a Taylor expansion in time and with all time derivatives replaced by spatial derivatives through the partial differential equations, resulting in a narrower effective stencil compared with previous high order finite difference WENO scheme based on the reconstruction of flux functions with a Lax-Wendroff time discretization. We will describe the scheme formulation and present numerical tests for one-and two-dimensional scalar and system conservation laws demonstrating the designed high order accuracy and non-oscillatory performance of the schemes constructed in this paper.

This paper presents a numerical scheme of arbitrary order of accuracy in both space and time, based on the arbitrary high-order derivatives methodology, for transient acoustic simulations. The scheme combines the nodal discontinuous Galerkin method for the spatial discretization and the Taylor series integrator (TSI) for the time integration. The main idea of the TSI is a temporal Taylor series expansion of all unknown acoustic variables in which the time derivatives are replaced by spatial derivatives via the Cauchy-Kovalewski procedure. The computational cost for the time integration is linearly proportional to the order of accuracy. To increase the computational efficiency for simulations involving strongly varying mesh sizes or material properties, a local time-stepping (LTS) algorithm accompanying the arbitrary high-order derivatives discontinuous Galerkin (ADER-DG) scheme, which ensures correct communications between domains with different time step sizes, is proposed. A numerical stability analysis in terms of the maximum allowable time step sizes is performed. Based on numerical convergence analysis, we demonstrate that for nonuniform meshes, a consistent high-order accuracy in space and time is achieved using ADER-DG with LTS. An application to the sound propagation across a transmissive noise barrier validates the potential of the proposed method for practical problems demanding high accuracy.

SUMMARY Inthispaper,weproposeatemporalfourth-orderschemeforsolvingthe2-Dfirst-orderacoustic wave equations with perfectly matched layers in time domain. For the temporal discretization, we approximate the velocity at mid-points and the pressure at grid points, then expand the velocity and pressure by Taylor series and replace the high-order temporal derivatives by spatial derivatives. For the spatial discretization, we apply the Fourier pseudospectral method based on conventional and staggered-grid schemes. The dispersion and stability of the new scheme are analysed, and numerical simulations are presented to confirm the robustness and efficiency of the scheme. The new scheme is eventually applied for the wavefield modelling and reverse time migration in bilayered and heterogeneous media to further verify its accuracy and efficiency.

Application of the Modern Taylor Series Method to a multi-torsion chain

To characterize a finite-difference time-domain (FDTD) scheme, the truncation error using Taylor's series and the numerical dispersion are often used. Truncation error analysis determines the order of accuracy, but cannot differentiate one scheme from another if they have the same order of accuracy. The theoretical relation for the numerical dispersion sometimes may be difficult to obtain. This paper introduces another way to characterize the error of an FDTD scheme quantitatively. This is the truncation error with plane wave propagation. The analytical expressions for such truncation errors for Yee 's FDTD and Crank-Nicolson FDTD are derived, and the errors are compared.

The Brinkman–Bénard convection problem is chosen for investigation, along with very general boundary conditions. Using the Maclaurin series, in this paper, we show that it is possible to perform a relatively exact linear stability analysis, as well as a weakly nonlinear stability analysis, as normally performed in the case of a classical free isothermal/free isothermal boundary combination. Starting from a classical linear stability analysis, we ultimately study the chaos in such systems, all conducted with great accuracy. The principle of exchange of stabilities is proven, and the critical Rayleigh number, Rac, and the wave number, ac, are obtained in closed form. An asymptotic analysis is performed, to obtain Rac for the case of adiabatic boundaries, for which ac≃0. A seemingly minimal representation yields a generalized Lorenz model for the general boundary condition used. The symmetry in the three Lorenz equations, their dissipative nature, energy-conserving nature, and bounded solution are observed for the considered general boundary condition. Thus, one may infer that, to obtain the results of various related problems, they can be handled in an integrated manner, and results can be obtained with great accuracy. The effect of increasing the values of the Biot numbers and/or slip Darcy numbers is to increase, not only the value of the critical Rayleigh number, but also the critical wave number. Extreme values of zero and infinity, when assigned to the Biot number, yield the results of an adiabatic and an isothermal boundary, respectively. Likewise, these extreme values assigned to the slip Darcy number yield the effects of free and rigid boundary conditions, respectively. Intermediate values of the Biot and slip Darcy numbers bridge the gap between the extreme cases. The effects of the Biot and slip Darcy numbers on the Hopf–Rayleigh number are, however, opposite to each other. In view of the known analogy between Bénard convection and Taylor–Couette flow in the linear regime, it is imperative that the results of the latter problem, viz., Brinkman–Taylor–Couette flow, become as well known.