Volume 48, pp. 329-347, 2018.

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

Miljenko Marušić

Abstract

We introduce a family of exponentially fitted difference schemes of arbitrary order as numerical approximations to the solution of a singularly perturbed two-point boundary value problem: $\varepsilon y” + b y' + c y = f$. The difference schemes are derived from interpolation formulae for exponential sums. The so-defined $k$-point differentiation formulae are exact for functions that are a linear 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 from the 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 of a 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 schemes in the case when $c<0$ and $b \neq 0$.

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Key words

difference scheme, singular perturbation, ODE, interpolation, exponential sum

AMS subject classifications

65L12, 65L11, 65L10, 65L20, 41A30, 65D25

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