2024 , Volume 29, № 3, p.70-80

Boundary and interior layers present serious difficulties for the efficient calculation of equations modelling many technical applications, in particular, those having a small parameter before the higher derivatives. Due to this phenomenon, developing uniformly convergent algorithms for solving such problems are difficult.

Resources provided by numerical schemes and adaptive grids can significantly reduce the adverse effects on the accuracy of numerical experiments due to the layers. An efficient and popular scheme for solving two-point singularly-perturbed problems with layers is the upwind difference scheme. However, this scheme provides convergence of the first order only.

In this paper, we are focused on two second-order uniformly convergent finite difference algorithms for solving two-point singularly-perturbed problems. The proposed algorithms apply a hybrid scheme based on the midpoint upwind approximation, Buleev’s scheme and special layer-resolving grids designed for solving problems with exponential and power layers of the first type.

Numerical experiments conducted out for singularly perturbed problems confirm the efficiency of the algorithms for various values of the small parameter and show that the proposed method provides competitive results compared to other methods available in the literature.