New classes of block backward differentiation formula for solving stiff initial value problems - Musa, Hamisu (2013)
Implicit numerical methods for solving sti® Initial Value Problems (IVPs) are known to perform better than explicit ones. There has been a great deal of in- terest to develop implicit block and non-block numerical methods for solving sti® IVPs. One of the most popular methods is the Backward Di®erentiation Formula (BDF). The BDF still remain a foundation for most widely used algorithms. In this thesis, new classes of block methods are developed for the solution of sti® initial value problems. The methods are based on the BDF and produce more than one solution value per step. The ¯rst class is a super class of the Block Backward Di®erentiation Formula (BBDF) and contains the BBDF as a subclass. This class has the advantage of generating di®erent set of formulae with A-stability properties by simply varying a value of a parameter within the interval (¡1; 1). 2¡point and 3¡point block methods of constant step size belonging to this class are developed and codes are designed to implement the methods. The stability analysis of the methods shows that they are A¡stable. The performance of the methods in terms of accuracy is seen to outperform the non¡block BDF and the BBDF methods of the same order. In addition, a 2¡point variable step size superclass of BBDF method is formulated. The strategy for controlling the step size ratio is described. The problems tested indicate the method's suitability for solving sti® IVPs. The second class of formulae developed involved the addition of an extra future point in the BBDF method to produce new formula called Block Extended Back- ward Di®erentiation Formula (BEBDF). 2¡point and 3¡point formulae of this class are also developed and their codes implemented. Using the same number of points, this class has the advantage of obtaining higher order A-stable methods than the BBDF. In addition, the accuracy is seen to be better than the BBDF. The stability of the methods developed is analyzed and the methods are found to possess A-stability properties. Both the codes developed proved to be e±cient for solving sti® initial value prob- lems.