Multiblock Backward Differentiation Formulae for solving first order ordinary differential equations - Mohd Nasir, Nor Ain Azeany (2011)
Block Backward Differentiation Formulae (BBDF) methods of variable order is derived to solve first order stiff ordinary differential equations (ODEs). These methods computed two approximate solutions yn+1 and yn+2 at the points xn+1 and xn+2 of the initial value problems (lVPs) concurrently in a block at each step. The numerical results are given to validate the method and the performancers are being compared with the classical Backward Differentiation Formulae (BBDF) methods Furthermore, the stability properties and the stability regions for the BBDF methods are investigated to ensure the methods are useful for solving stiff ODEs. This BBDF is extended to variable order method in order to improve the efficiency of the method. A single code is developed with fixed stepsize and implemented using Microsoft Visual C++ 2008 Express Edition and compared with ode 15s and ode 23s which is run in MATLAB 7.1. A parallel scheme of the BBDF is derived in order to solve large problems in ODEs using the Message Passing Interface (MPI) library run by the High Performance Computer (HPC). The efficiency of the method is justified by the numerical results given. The results generated showed that these methods produced less computional time and achieved the desired accuracy.