The stabilization of high-order multistep schemes for the Laguerre one-way wave equation solver

Abstract

This paper considers spectral-difference methods of a high-order of accuracy for solving the one-way wave equation using the Laguerre integral transform with respect to time as the base. In order to provide a high spatial accuracy and calculation stability, the Richardson method can be employed. However such an approach requires high computer costs, therefore we consider alternative algorithms based on the Adams multistep schemes. To reach the stability first for the 1D one-way equation and then for the 2D case, the stabilizing procedures using the spline interpolation were developed. This made possible to efficiently implement a predictor-corrector type method in terms of which a boundary value problem for highorder elliptic equations is substituted for a sequence of inversions of second order elliptic operators thus decreasing computer costs. To assess the accuracy and stability of difference approximations for the 1D one-way wave equation, the analytical solution based on the double Laguerre transform was obtained. This solution can be efficiently calculated if for summation one makes use of fast algorithms of computing a discrete linear convolution. For the 2D one-way wave equation the stability and accuracy of the procedures proposed have been studied on implementing the migration algorithm within a problem of seismic prospecting.

A three-dimensional Laguerre one-way wave equation solver

“…The Laguerre method is based on a real Padé approximation (2), which may cause instability of the calculations. In the previous versions of the algorithm, the calculations were stabilized by procedures of Richardson extrapolation [27] or spline filtering [28]. However, this approach imposes additional restrictions on the integration step in depth, ∆z.…”

“…Unfortunately, for the frequencies ω < ω 0 the matrices will be ill-conditioned, which significantly decreases the efficiency of the Fourier method for solving three-dimensional problems. To decrease the computational work, it is proposed in papers [27,28] to use (instead of the Fourier transform in time) the Laguerre transform…”

Generating Laguerre expansion coefficients by solving a one-dimensional transport equation

“…However, n/p expansion coefficients may be insufficient to approximate a nonsmooth local function, which results in loss in approximation accuracy. Nevertheless, if one has to approximate a time series with an accuracy of the order of ǫ = 10 −3 ÷ 10 −5 (which is sufficient for practical calculations [8,9]), it is recommended to use algorithm 4.…”

“…The Laguerre integral transform has been used in various fields of mathematical simulation to solve acoustics and elasticity equations [1,2,3,4], Maxwell and heat conduction equations [5,6], and spectroscopy problems [7]. The Laguerre transform has proved to be a very efficient tool in constructing a stable algorithm of wave field continuation when solving inverse problems of seismic prospecting [8,9] and many others. The Laguerre transform has served as a basis for the development of numerical methods of inversion of Laplace [10,11,12] and Fourier [13] transforms.…”