2025 , Volume 30, № 3, p.62-77

This paper presents a method of constructing compact difference schemes for the Ginzburg - Landau equation for both plane and spatial problems in nonlinear fiber optics with central symmetry. The method is based on the formation of special difference analogs for “complex second derivative” operators using stensil functionals. The peculiarity of the method is in the form of the main error term, which has the form of the square of the initial operator with the accuracy up to the multiplier, as in the case of the usual second derivative. With this approach, the technology of constructing compact schemes in orthogonal curvilinear coordinates becomes quite similar to the traditional technology for Cartesian coordinates. As a result, rather universal approach to the construction of compact difference schemes for the Ginzburg–Landau equation, actually applicable for solving problems in various orthogonal curvilinear coordinate systems.

The question of setting the boundary conditions in poles of spherical and polar coordinate systems is discussed. The basis of the boundary condition is a difference scheme in Cartesian coordinates, written in the pole, with 5-point approximations of second derivatives of the fourth order of accuracy. In the symmetric case this approach leads to a three-point boundary condition, and in asymmetric problems the solution in the pole is determined using runs along the Cartesian semi-axes in the direction from the periphery to the center.

Numerical experiments on a sequence of densifying grids have been performed on test problems. Aposteriori error estimates and estimates of real orders of accuracy were obtained. The calculations showed a significant advantage of compact schemes over the widely used Crank–Nicholson scheme, as well as a satisfactory coincidence of the real orders of accuracy with the theoretically expected ones.