2026 , Volume 31, № 2, p.76-89

The paper addresses the development of numerical algorithms based on layer-resolving grids and special hybrid difference schemes for solving one-dimensional stationary problems with a small diffusion coefficient that have boundary and interior layers.

It describes coordinate transformations and corresponding layer-resolving grids, and analyses the convergence of numerical solutions by using the hybrid schemes on the layer-resolving grids. Theoretical results are confirmed by numerical experiments.

The layers are classified by the qualitative behaviour of the corresponding singular functions, in particular by their derivatives. At present, basic types of layers (exponential, power-of-type-1 and type-2, and logarithmic) and their combinations (hybrid types of layers) are known for solving problems with small parameters. Of course, solutions of real problems, in particular to the Navier – Stokes equations, may have new types of layers that have not yet been discovered.

The relevance and novelty of the paper are concluded on two powerful factors for increasing efficiencies of algorithms, namely, adaptive grids with special rules for grid clustering in the zones of layers which provide uniform convergence of numerical solutions and hybrid difference schemes. Synthesis of the hybrid schemes and special meshes constructed using a priori information about the behaviour of solutions in layers and eliminating layers in transformed coordinates allows achieving uniform convergence to the second order and to increase the real accuracy of approximate solutions practically without complicating the algorithms typical of homogeneous schemes of the up-wind type.

The developed technologies of generation of special meshes and construction of difference schemes can be applied for numerical solution of problems with boundary and internal layers of various types, including modelling of viscous fluid and gas flows.