Fourth-order finite-difference methods for two-point boundary-value problems

A number of new fourth-order accurate finite-difference methods are developed for second-order ordinary differential equations of the boundary-value type. Schemes are obtained for both linear and non-linear equations. In all cases, the solution of the difference equations may be accomplished using a direct elimination technique for linear tridiagonal matrix problems. The accuracy of the new methods is compared with existing finite-difference methods on a theoretical basis as well as by considering a number of example problems. It is concluded that the new methods offer significant advantages for specific types of equations in terms of accuracy and/or computational efficiency.