Derivation of a Class of Generalized Adams Methods

In this book, we derived the continuous form of a class of block Generalized Adams Methods for step numbers k=4 and 8 with continuous coefficients based on multistep collocation using the matrix inverse collocation approach. In this research work, additional equations are gotten from the continuous formulation of the Generalized Adams method for each step number k. These new additional equations and the discrete schemes derived by Brugnano and Trigiante shall be recovered from the continuous formulation of a class of Generalized Adams Methods for each step number. The convergence and order of the new derived schemes were analysed and the stability regions of the block method were plotted. The collocation technique yielded A-stable 4 and 8 step GAMs. The new schemes were of uniform order. The numerical solutions of Stiff and non-linear IVPs shows that these classes of methods are good for the solution of Stiff ODEs and performed relatively better upon comparison with the conventional Generalized Adams Methods for step number k=4 and for step number k=8. Numerical examples obtained demonstrate the accuracy and efficiency of the new block methods.