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

## TL;DR

This paper develops stable, high-order spectral-difference methods for solving the one-way wave equation using Laguerre transforms, employing spline interpolation for stability and predictor-corrector algorithms to reduce computational costs.

## Contribution

It introduces stabilizing procedures based on spline interpolation for high-order multistep schemes solving the Laguerre one-way wave equation, improving stability and efficiency.

## Key findings

- Stable high-order schemes achieved for 1D and 2D wave equations.
- Efficient predictor-corrector method reduces computational costs.
- Validated accuracy and stability through seismic migration applications.

## 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 high-order 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.