Exponentially-fitted methods on layer-adapted meshes

In this paper, a new derivation of a uniformly-convergent, second-order method for singularly-perturbed, linear ordinary differential equations based on the freezing of the coefficients of the differential equation, and integration of the resulting equations subject to continuity and smoothness conditions at the nodes, is presented. The derivation presented here is compared with others based on Green's functions, when only advection and diffusion processes are considered when solving the homogeneous equations. In addition, a new method that accounts for advection, diffusion and reaction processes when solving the homogeneous equation is also presented. The two exponentially-fitted techniques presented in the paper are used on layer-adapted meshes which are piecewise uniform and concentrate grid points in the boundary layers, and their results are compared with those obtained with upwind methods in piecewise-uniform meshes. It is shown that standard techniques on piecewise-uniform meshes are less accurate than exponentially-fitted ones, and the accuracy of the latter may not improve by employing layer-adapted piecewise-uniform meshes due to the large change in the step size at the transition points. The paper also presents an exponentially-fitted method for singularly-perturbed, periodic, two-point boundary-value problems of ordinary differential equations.