Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements

The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical Methods for Singularly Perturbed Differential Equations" appeared many years ago and was for many years a reliable guide into the world of numerical methods for singularly perturbed problems. Since then many new results came into the game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827

An Upwind Finite Difference Method to Singularly Perturbed Convection Diffusion Problems on a Shishkin Mesh

This paper introduces a numerical approach to solve singularly perturbed convection diffusion boundary value problems for second-order ordinary differential equations that feature a small positive parameter {\epsilon} multiplying the highest derivative. We specifically examine Dirichlet boundary conditions. To solve this differential equation, we propose an upwind finite difference method and incorporate the Shishkin mesh scheme to capture the solution near boundary layers. Our solver is both direct and of high accuracy, with computation time that scales linearly with the number of grid points. MATLAB code of the numerical recipe is made publicly available. We present numerical results to validate the theoretical results and assess the accuracy of our method. The tables and graphs included in this paper demonstrate the numerical outcomes, which indicate that our proposed method offers a highly accurate approximation of the exact solution.Comment: 19 pages, 4 figures. arXiv admin note: text overlap with arXiv:2305.18711 by other author