Two new compact finite-difference schemes for the solution of boundary value problems in second-order non-linear ordinary differential equations, using non-uniform grids

Abstract

Two new finite-difference, three-point compact discretisations of boundary value problems in second-order non-linear ordinary differential equations: $U^{(2)}$(x)-F(x, U(x), $U^{(1)}$(x))=0, applicable to non-uniform grids, are derived and compared in calculations with three previously published methods. The discretisations are modifications of the efficient uniform-grid arithmetic average method of Chawla and Shivakumar [Neural Parallel Sci. Comput. 4 (1996) 387-396]. Formal error analysis reveals that the truncation errors of the new discretisations are of third order with respect to the local grid spacings. New, economical, fourth-order accurate, two-point compact approximations to the first derivatives of the solution at the boundaries are also designed. Numerical experiments with the solution of singularly perturbed example problems, characterized by boundary and interior layers, indicate that the practical orders of accuracy of the new schemes are close to four, even for non-uniform grids. The methods prove competitive with the three-point extension of the Numerov method, and with one other extension of the Chawla and Shivakumar method.