Two-way Splitting Research Articles

Solving the spin-2 Gross-Pitaevskii equation using exact nonlinear dynamics and symplectic composition.

We develop numerical methods for solving the spin-2 Gross-Pitaevskii equation. The basis of our work is a two-way splitting of this evolution equation that leads to two exactly solvable subsystems. Utilizing second-order and fourth-order composition schemes we realize two fully symplectic integration algorithms, the first such algorithms for evolving spin-2 condensates. We demonstrate the accuracy of these algorithms against other methods on application to an exact continuous wave solution that we derive.

Efficient and accurate methods for solving the time-dependent spin-1 Gross-Pitaevskii equation.

We develop a numerical method for solving the spin-1 Gross-Pitaevskii equation. The basis of our work is a two-way splitting of the spin-1 evolution equation that leads to two exactly solvable flows. We use this to implement a second-order and a fourth-order symplectic integration method. These are the first fully symplectic methods for evolving spin-1 condensates. We develop two nontrivial numerical tests to compare our methods against two other approaches.

Reducing two-way splitting error of FFD method in dual domains

The Fourier finite-difference (FFD) method is very popular in seismic depth migration. But its straightforward 3D extension creates two-way splitting error due to ignoring the cross terms of spatial partial derivatives. Traditional correction schemes, either in the spatial domain by the implicit finite-difference method or in the wavenumber domain by phase compensation, lead to substantially increased computational costs or numerical difficulties for strong velocity contrasts. We propose compensating the two-way splitting error in dual domains, alternately in the spatial and wavenumber domains via Fourier transform. First, we organize the expanded square-root operator in terms of two-way splitting FFD plus the usually ignored cross terms. Second, we select a group of optimized coefficients to maximize the accuracy of propagation in both inline and crossline directions without yet considering the diagonal directions. Finally, we further optimize the constant coefficient of the compensation part to further improve the overall accuracy of the operator. In implementation, the compensation terms are similar to the high-order corrections of the generalized-screen method, but their functions are to compensate the two-way splitting error rather than the expansion error. Numerical experiments show that optimized one-term compensation can achieve nearly perfect circular impulse responses and the propagation angle with less than 1% error for all azimuths is improved up to 60° from 35°. Compared with traditional single-domain methods, our scheme can handle lateral velocity variations (even for strong velocity contrasts) much more easily with only one additional Fourier transform based on the two-way splitting FFD method, which helps retain the computational efficiency.

A Comparison of Splitting Techniques for 3D Complex Padé Fourier Finite Difference Migration

Three-dimensional wave-equation migration techniques are still quite expensive because of the huge matrices that need to be inverted. Several techniques have been proposed to reduce this cost by splitting the full 3D problem into a sequence of 2D problems. We compare the performance of splitting techniques for stable 3D Fourier finite-difference (FFD) migration techniques in terms of image quality and computational cost. The FFD methods are complex Padé FFD and FFD plus interpolation, and the compared splitting techniques are two- and four-way splitting as well as alternating four-way splitting, that is, splitting into the coordinate directions at one depth and the diagonal directions at the next depth level. From numerical examples in homogeneous and inhomogeneous media, we conclude that, though theoretically less accurate, alternate four-way splitting yields results of comparable quality as full four-way splitting at the cost of two-way splitting.

Optimized Chebyshev Fourier migration: A wide-angle dual-domain method for media with strong velocity contrasts

A wide-angle propagator is essential when imaging complex media with strong lateral velocity contrasts in one-way wave-equation migration. We have developed a dual-domain one-way propagator using truncated Chebyshev polynomials and a globally optimized scheme. Our method increases the accuracy of the expanded square-root operator by adding two high-order terms to the traditional split-step Fourier propagator. First, we approximate the square-root operator using Taylor expansion around the reference background velocity. Then, we apply the first-kind Chebyshev polynomials to economize the results of the Taylor expansion. Finally, we optimize the constant coefficients using the globally optimized scheme, which are fixed and feasible for arbitrary velocity models. Theoretical analysis and nu-merical experiments have demonstrated that the method has veryhigh accuracy and exceeds the unoptimized Fourier finite-difference propagator for the entire range of practical velocity contrasts. The accurate propagation angle of the method is always about 60° under the relative error of 1% for complex media with weak, moderate, and even strong lateral velocity contrasts. The method allows us to handle wide-angle propagations and strong lateral velocity contrast simultaneously by using Fourier transform alone. Only four 2D Fourier transforms are required for each step of 3D wavefield extrapolation, and the computing cost is similar to that of the Fourier finite-difference method. Compared with the finite-difference method, our method has no two-way splitting error (i.e., azimuthal-anisotropy error) for 3D cases and almost no numerical dispersion for coarse grids. In addition, it has strong potential to be accelerated when an enhanced fast Fourier transform algorithm emerges.

Modeling 3‐D Scalar Waves Using the Fourier Finite‐Difference Method

AbstractFourier finite‐difference (FFD) operator has both of the advantages of Fourier and finite‐difference methods, it can handle the wave propagations in complex media with large velocity contrasts and wide propagation angles. In 3‐D case, however, two‐way splitting may cause artificial azimuthal anisotropy. In this paper, we handle the remnants of the conventional two‐way splitting technique by Fourier transforms, and the azimuthal anisotropy is removed. Based on the dual‐domain scheme of the FFD method, the proposed algorithm significantly reduces computational cost caused by error correction. Compared with the finite‐difference method by the zero‐offset records and shot profiles based on 3‐D modified French model, the proposed method can properly model the primary reflected waves and is much more attractive in both computational cost and storage demand.