Numerical solution of singularly perturbed boundary value problems using a Shishkin mesh based on spline-in-compression approximations

A class of singularly perturbed convection–diffusion boundary value problems for second-order ordinary differential equations without interior turning points, involving a small perturbation parameter multiplying the highest-order derivative and subject to Dirichlet boundary conditions, is solved using a Shishkin mesh based on spline-in-compression. The proposed method effectively captures the boundary layer behavior and ensures improved numerical accuracy. To demonstrate the efficiency and robustness of the scheme, two numerical examples are presented and the results are compared with those obtained by using variable-mesh and uniform mesh approaches. The comparative analysis confirms the superior performance of the proposed method, particularly for problems exhibiting sharp boundary layers.