A stability formula for Lax-Wendroff methods with fourth-order in time and general-order in space for the scalar wave equation

Geophysics ◽  
2011 ◽  
Vol 76 (2) ◽  
pp. T37-T42 ◽  
Author(s):  
Jing-Bo Chen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Fourth Order ◽  
Second Order ◽  
High Order ◽  
Scalar Wave ◽  
Scalar Wave Equation ◽  
General Order ◽  
Second Order In Time ◽  
Spatial Derivatives

Based on the formula for stability of finite-difference methods with second-order in time and general-order in space for the scalar wave equation, I obtain a stability formula for Lax-Wendroff methods with fourth-order in time and general-order in space. Unlike the formula for methods with second-order in time, this formula depends on two parameters: one parameter is related to the weights for approximations of second spatial derivatives; the other parameter is related to the weights for approximations of fourth spatial derivatives. When discretizing the mixed derivatives properly, the formula can be generalized to the case where the spacings in different directions are different. This formula can be useful in high-accuracy seismic modeling using the scalar wave equation on rectangular grids, which involves both high-order spatial discretizations and high-order temporal approximations. I also prove the instability of methods obtained by applying high-order finite-difference approximations directly to the second temporal derivative, and this result solves the “Bording’s conjecture.”

Modeling the scalar wave equation with Nyström methods

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. T151-T158 ◽  
Author(s):  
Jing-Bo Chen
Keyword(s):  
Wave Equation ◽  
Fourth Order ◽  
Second Order ◽  
Scalar Wave ◽  
Nyström Method ◽  
Scalar Wave Equation ◽  
Nystrom Method ◽  
Nyström Methods ◽  
Long Time ◽  
Order Scheme

High-accuracy numerical schemes for modeling of the scalar wave equation based on Nyström methods are developed in this paper. Space is discretized by using the pseudospectral algorithm. For the time discretization, Nyström methods are used. A fourth-order symplectic Nyström method with pseudospectral spatial discretization is presented. This scheme is compared with a commonly used second-order scheme and a fourth-order nonsymplectic Nyström method. For a typical time-step size, the second-order scheme exhibits spatial dispersion errors for long-time simulations, while both fourth-order schemes do not suffer from these errors. Numerical comparisons show that the fourth-order symplectic algorithm is more accurate than the fourth-order nonsymplectic one. The capability of the symplectic Nyström method in approximately preserving the discrete energy for long-time simulations is also demonstrated.

scholarly journals An Efficient Compact Difference Method for Temporal Fractional Subdiffusion Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Lei Ren ◽  
Lei Liu
Keyword(s):  
Finite Difference ◽  
Fourth Order ◽  
Difference Method ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Approximation Formula ◽  
Discrete Energy ◽  
Compact Finite Difference ◽  
Second Order In Time

In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain 2-α in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the L2-1σ approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.

Temporal high-order staggered-grid finite-difference schemes for elastic wave propagation

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. T207-T224 ◽  
Author(s):  
Zhiming Ren ◽  
Zhen Chun Li
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Finite Difference ◽  
Elastic Wave ◽  
High Order ◽  
Elastic Wave Propagation ◽  
Staggered Grid ◽  
S Wave ◽  
Order Accuracy ◽  
Spatial Derivatives

The traditional high-order finite-difference (FD) methods approximate the spatial derivatives to arbitrary even-order accuracy, whereas the time discretization is still of second-order accuracy. Temporal high-order FD methods can improve the accuracy in time greatly. However, the present methods are designed mainly based on the acoustic wave equation instead of elastic approximation. We have developed two temporal high-order staggered-grid FD (SFD) schemes for modeling elastic wave propagation. A new stencil containing the points on the axis and a few off-axial points is introduced to approximate the spatial derivatives. We derive the dispersion relations of the elastic wave equation based on the new stencil, and we estimate FD coefficients by the Taylor series expansion (TE). The TE-based scheme can achieve ([Formula: see text])th-order spatial and ([Formula: see text])th-order temporal accuracy ([Formula: see text]). We further optimize the coefficients of FD operators using a combination of TE and least squares (LS). The FD coefficients at the off-axial and axial points are computed by TE and LS, respectively. To obtain accurate P-, S-, and converted waves, we extend the wavefield decomposition into the temporal high-order SFD schemes. In our modeling, P- and S-wave separation is implemented and P- and S-wavefields are propagated by P- and S-wave dispersion-relation-based FD operators, respectively. We compare our schemes with the conventional SFD method. Numerical examples demonstrate that our TE-based and TE + LS-based schemes have greater accuracy in time and better stability than the conventional method. Moreover, the TE + LS-based scheme is superior to the TE-based scheme in suppressing the spatial dispersion. Owing to the high accuracy in the time and space domains, our new SFD schemes allow for larger time steps and shorter operator lengths, which can improve the computational efficiency.

scholarly journals Second-order scalar wave field modeling with a first-order perfectly matched layer

Solid Earth ◽  
2018 ◽  
Vol 9 (6) ◽  
pp. 1277-1298
Author(s):  
Xiaoyu Zhang ◽  
Dong Zhang ◽  
Qiong Chen ◽  
Yan Yang
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Forward Modeling ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Staggered Grid ◽  
Scalar Wave ◽  
First Order ◽  
Second Order Wave Equation ◽  
Boundary Absorption

Abstract. The forward modeling of a scalar wave equation plays an important role in the numerical geophysical computations. The finite-difference algorithm in the form of a second-order wave equation is one of the commonly used forward numerical algorithms. This algorithm is simple and is easy to implement based on the conventional grid. In order to ensure the accuracy of the calculation, absorption layers should be introduced around the computational area to suppress the wave reflection caused by the artificial boundary. For boundary absorption conditions, a perfectly matched layer is one of the most effective algorithms. However, the traditional perfectly matched layer algorithm is calculated using a staggered grid based on the first-order wave equation, which is difficult to directly integrate into a conventional-grid finite-difference algorithm based on the second-order wave equation. Although a perfectly matched layer algorithm based on the second-order equation can be derived, the formula is rather complex and intermediate variables need to be introduced, which makes it hard to implement. In this paper, we present a simple and efficient algorithm to match the variables at the boundaries between the computational area and the absorbing boundary area. This new boundary-matched method can integrate the traditional staggered-grid perfectly matched layer algorithm and the conventional-grid finite-difference algorithm without formula transformations, and it can ensure the accuracy of finite-difference forward modeling in the computational area. In order to verify the validity of our method, we used several models to carry out numerical simulation experiments. The comparison between the simulation results of our new boundary-matched algorithm and other boundary absorption algorithms shows that our proposed method suppresses the reflection of the artificial boundaries better and has a higher computational efficiency.

High-order doubly asymptotic open boundaries for scalar wave equation

2009 ◽  
Vol 79 (3) ◽  
pp. 340-374 ◽  
Author(s):  
Suriyon Prempramote ◽  
Chongmin Song ◽  
Francis Tin-Loi ◽  
Gao Lin
Keyword(s):  
Wave Equation ◽  
High Order ◽  
Scalar Wave ◽  
Scalar Wave Equation ◽  
Open Boundaries
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