Two-Step Second Derivative Methods with Hermite Polynomial as Basis Function for Solving Stiff Problems

Abstract

In this paper, a class of two-step second derivative numerical method is developed by incorporating one or more function evaluation at collocation with carefully selected off-grid points. The continuous formulations of the methods are derived through the interpolation and collocation technique with Hermite polynomial as basis function. The two numerical schemes derived are of higher order of accuracy with relatively small error constants. The methods are consistent and zero stable and hence convergent. The stability properties of the methods are carried out via the linear system. Both methods are A-stable as their regions of absolute stability contain the entire left-hand plane of the stability region. Furthermore, the two methods were implemented as block forms in other to simultaneously produce approximate solutions to some standard stiff problems (both linear and nonlinear) found in the literature. Hence our methods are self-starting and do not require separate methods to start the implementation. The errors incurred in our methods on the problems considered, are relatively lower than the methods found in the literature.