Adaptive Enriched Rational Spectral Methods with sinh transformations and asymptotic correctors for variable-coefficient singular perturbation problems

This paper introduces and analyzes novel Enriched Rational Spectral Methods for efficiently solving singular perturbation problems exhibiting sharp boundary layers. While spectral methods are known for their ‘spectral accuracy’ in solving smooth problems, their performance deteriorates for stiff differential equations because they fail to resolve rapid transitions in the solution. To overcome this limitation, we propose a rational spectral collocation framework enriched with asymptotic corrector functions. These correctors are derived directly from a boundary layer analysis of the variable-coefficient operator itself, enabling them to accurately capture the solution's singular behavior. Two specific schemes are proposed: the Enriched Spectral Method (ESM) and the Enriched Rational Spectral Method combined with a sinh transformation (ERSM-sinh). In ERSM-sinh, the corrector functions are integrated with a sinh transformation whose parameters—layer location and width—are determined from asymptotic estimates. The correction parameters are obtained implicitly by solving the discrete algebraic system arising from the original problem. Extensive numerical experiments on convection-diffusion and reaction-diffusion problems with variable coefficients demonstrate the superior performance of our methods. Results show that ERSM-sinh maintains robust spectral accuracy, significantly outperforms existing approaches such as RSC-SSM and RSCAT for variable-coefficient problems, and achieves high precision with minimal computational cost—even for very small perturbation parameters (e.g., ε = 10−10). This work provides a high-resolution, efficient, and generalizable framework for singularly perturbed boundary value problems.