Direct two-point block methods for solving nonstiff higher order Ordinary Differential Equations using Backward Difference formulation - Mohd Ijam, Hazizah (2014)
This thesis describes the development of a Two-Point Block Backward Difference method (2PBBD) for solving system of nonstiff higher order Ordinary Differential Equations (ODEs) directly. The method computes the approximate solutions; and at two points and simultaneously within an equidistant block. This method has the advantages of calculating the integration coefficients only once at the beginning of the integration. The relationship between the explicit and implicit coefficients has also been derived. These motivate us to formulate the association between the formula for predictor and corrector. The relationship between the lower and higher order derivative also have been established. New explicit and implicit block methods using constant step sizes and three back values have also been derived. The algorithm developed is implemented using Microsoft Visual C++ 6.0 and run by High Performance Computer (HPC) using the Message Passing Interface (MPI) library. The stability properties for the 2PBBD methods are analyzed to ensure its suitability for solving nonstiff Initial Value Problems (IVPs). The stability analysis shows that the method is stable.Numerical results are presented to compare the performances of this method with the previously published One-Point Backward Difference (1PBD) and Two-Point Block Divided Difference (2PBDD) methods. The numerical results indicated that for finer step sizes, 2PBBD performs better than 1PBD and 2PBDD.