High-order exponentially fitted difference schemes for singularly perturbed two-point boundary value problems

Abstract

We introduce a family of exponentially fitted difference schemes of arbitrary orderas numerical approximations to the solution of a singularly perturbed two-point boundary valueproblem: $\\varepsilon y” + b y\' + c y = f$.The difference schemes are derived from interpolation formulae for exponential sums.The so-defined $k$-point differentiation formulae are exact for functions that are alinear combination of $1,x,\\ldots,x^{k-2},\\exp{(-\\rho x)}$.The parameter $\\rho$ is chosen from the asymptotic behavior of the solution in the boundary layer.This approach allows a construction of the method with arbitrary order of consistency.Using an estimate for the interpolation error, we prove consistency of all the schemes fromthe family.The truncation error is bounded by $C h^{k-2}$, where $C$ is a constant independent of $\\varepsilon$and $h$.Therefore, the order of consistency for the $k$-point scheme is $k-2$ ($k \\geq 3$) in case ofa small perturbation parameter $\\varepsilon$.There is no general proof of stability for the proposed schemes.Each scheme has to be considered separately.In the paper, stability, and therefore convergence, is proved for three-point schemesin the case when $c<0$ and $b \\neq 0$.