2024 , Volume 29, № 5, p.43-54

The paper is a further development of the authors’ research cycle on high-precision difference methods for numerical solution of nonlinear fiber optics problems (for the Schr¨odinger equation and the Ginzburg – Landau equation). A two-layer predictor-corrector type scheme is chosen as the apparatus for computations on non-uniform meshes, which is constructed by analogy with the wellproven noniterative scheme previously proposed by the authors for the special case of a uniform grid. In contrast to the frequently used three-layer scheme, which requires setting initial conditions on the first two layers on the evolutionary variable, the two-layer technique has no problems when a computation is started.

Instead of the frequently used method of adaptive mesh generation relying on the construction of a control function based on a dynamically varying solution, an explicit method of setting the denser mesh in the zones of large gradients, based on the classical basic Bakhvalov transformation and on the knowledge of soliton locations, is applied. The global coordinate transformation is constructed by smooth conjugation of elementary basic transformations. It allows saving computational resources and avoiding undesirable larger grid step near extrema of the second derivative of the solution, for example, near the soliton peak or base.

On two test problems the comparison of schemes of the fourth and second order of accuracy on constructed special grids and on uniform grids with the same number of steps is carried out. The order of accuracy of the schemes at moderate gradients of the solution is numerically confirmed. It is shown that at large gradients a reasonably constructed adaptive grid is a more important factor of computational efficiency than the order of accuracy of the schemes.