A continuous two-step method of order 8 with a block extension for y″=f(x,y,y′)y″=f(x,y,y′)

We derive a piecewise continuous hybrid third derivative approximation (CHTDA) that is defined for all values of the independent variable on the range of interest. This continuous approximation has the ability to provide a continuous solution between all the grid points with a uniform accuracy comparable to that obtained at the grid points. Furthermore, the CHTDA is used to produce several implicit hybrid third derivative formulas (IHTDFs), which are expressed in block form and applied as a block hybrid third derivative algorithm (BHTDA) for the numerical solution of second order initial value problems. The stability of the block extension is discussed. Numerical examples are given to illustrate the accuracy and efficiency of the BHTDA.