Stable high order methods for differential equations with small coefficients for the second order terms.

*(English)*Zbl 0770.65049A two-point boundary value problem with a small coefficient in the second derivative term is considered. Second-, fourth-, and sixth-order difference methods are constructed by adding to the conventional second- order centered scheme some terms obtained via the differentiation of the governing equation and the discretization of high-order derivatives.

The resulting schemes with tridiagonal or pentadiagonal difference operators are shown to be stable in the maximum norm if some conditions on the mesh size and the coefficients of the difference equations are imposed. Error estimates are presented which indicate at least second order convergence to the exact solutions independently of the small coefficient provided that these solutions do not depend on this coefficient.

To deal with the boundary-layer type of the exact solution, it is recommended to refine the mesh in the boundary layer region by a suitable mapping. Numerical examples are presented which illustrate the theoretical conclusions.

The resulting schemes with tridiagonal or pentadiagonal difference operators are shown to be stable in the maximum norm if some conditions on the mesh size and the coefficients of the difference equations are imposed. Error estimates are presented which indicate at least second order convergence to the exact solutions independently of the small coefficient provided that these solutions do not depend on this coefficient.

To deal with the boundary-layer type of the exact solution, it is recommended to refine the mesh in the boundary layer region by a suitable mapping. Numerical examples are presented which illustrate the theoretical conclusions.

Reviewer: A.I.Tolstykh (Moskva)

##### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

65L70 | Error bounds for numerical methods for ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

##### Keywords:

two-point boundary value problem; difference methods; Error estimates; second order convergence; boundary layer; Numerical examples
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\textit{J. Y. Choo} and \textit{D. H. Schultz}, Comput. Math. Appl. 25, No. 1, 105--123 (1993; Zbl 0770.65049)

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