Summary
The extended Numerov scheme of Chawla, adopted for nonuniform grids, is a useful compact finite-difference discretisation, suitable for the numerical solution of boundary value problems in singularly perturbed second order non-linear ordinary differential equations. A new set of three-point compact approximations to first and second derivatives, related to the Chawla scheme and valid for nonuniform grids, is developed in the present work. The approximations economically re-use intermediate quantities occurring in the Chawla scheme. The theoretical orders of accuracy are equal four for the central and one-sided first derivative approximations obtained, whereas the central second derivative formula is either fourth, third, or second order accurate, depending on the grid ratio. The approximations can be used for accurate a posteriori derivative evaluations. A Hermitian interpolation polynomial, consistent with the derivative approximations, is also derived. The values of the polynomial can be used, among other things, for guiding adaptive grid refinement. Accuracy orders of the new derivative approximations, and of the interpolating polynomial, are verified by computational experiments.
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Bieniasz, L.K. A set of compact finite-difference approximations to first and second derivatives, related to the extended Numerov method of Chawla on nonuniform grids. Computing 81, 77–89 (2007). https://doi.org/10.1007/s00607-007-0239-x
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DOI: https://doi.org/10.1007/s00607-007-0239-x