Block backward differentiation formula for solving ordinary and algebraic differential equations - Abasi, Naghmeh (2014)
This research focuses on solving semi-explicit index-1 Di®erential Algebraic Equations (DAEs) which is a special case of Di®erential Algebraic Equations (DAEs). Block Backward Di®erentiation Formula (BDF) methods of constant and variable step sizes are considered to produce more than one solutions per step for the DAEs concurrently. A formula of the 2-point with o®-step points using block BDF method of constant step size for solving sti® ODEs is developed. The stability analysis shows that the method is A-stable. The method has competitive results in comparison with the existing block BDF method in terms of accuracy and time. The 2-point, 3-point and 2-point with o®-step points block backward di®erentiation formulae of constant step size are extended for solving semi-explicit index-1 Di®erential Algebraic Equations (DAEs). Newton's iteration is used for the implementation of the methods. It is seen that the block BDF methods applied are more suitable than the existing BDF method in terms of accuracy and the time is competitive. In addition, a 3-point block backward di®erentiation formula using variable step size for solving sti® Ordinary Di®erential Equations (ODEs) is for mulated. The strategy applied for selecting the step size and the stability regions are described. The accuracy of the developed method is seen to be better than the existing variable step block BDF. Solving semi-explicit index-1 DAEs using 2-point and 3-point block backward di®erentiation formula of variable step size are also considered. The strategies involved in the choosing and controlling the step size of both methods are described. The codes developed indicate that the methods have outperformed the existing method in reducing the error while the time is competitive. The numerical results indicate that the block BDF methods of constant and variable step size for solving semi-explicit index-1 DAEs have better accuracy and e±ciency in comparison with the existing constant and variable step BDF methods.