engleski

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

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 the interpolation formulae for exponential sum. So defined $k$-point differentiation formulae are exact on the functions that are linear combination of $1, x, \ldots, x^{; ; ; k- 2}; ; ; , \exp{; ; ; (-\rho x)}; ; ; $. Parameter $\rho$ is chosen from the asymptotic behavior of the solution in the boundary layer. This approach makes possible a construction of the method of arbitrary order of consistency. Using an estimate for interpolation error, we prove consistency of all the schemes from the family. Truncation error is bounded by $C h^{; ; ; k-2}; ; ; $ where $C$ is a constant independent on $\varepsilon$ and $h$. Therefore, order of consistency for $k$ point scheme is $k-2$ ($k \geq 3$) in the case of small perturbation parameter $\varepsilon$. There is no general proof for stability of 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 $b \neq 0$.

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

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