Adaptive step size of diagonally implicit block backward differentiation formulas for solving first and second order stiff ordinary differential equations with applications

In this thesis, new classes of block methods based on backward differentiation formula (BDF) for solving first and second order stiff ordinary differential equations (ODEs) are developed. These methods are implemented in diagonally implicit structure and generated the solutions of yn+1 and yn+2 simultaneously in a block. The formulas are constructed by taking a non zero arbitrary, incorporating a free parameter, ρ and hence producing ρ diagonally implicit block backward differ- entiation formula (ρ DIBBDF) which contain the block backward differentiation formula (BBDF) as a subclass. Initially, the derivation of ρ DIBBDF in fixed and adaptive step approaches for the solution of first order stiff ODEs have been described. The classes have the advantage of producing a different set of formulas that possess A-stability properties by selecting the ρ value within the interval (-1,1). The order, consistency, zero stability, absolute stability and stability region for the methods have been determined to ensure their applicability in solving the stiff ODEs. The numerical results have marginally better performance for the fixed step formula and competitive achieve- ment for the adaptive step formula when compare to the existing BBDF methods. To deal with the system of second order stiff ODEs, ρ DIBBDF is formulated suited well with the systems in its original form, without transforms it to the first order ODEs. The convergence and stability properties also have been analyzed. ls for the method have been obtained and their stability regions have been discussed. The methods are implemented in fixed and adaptive step approaches. Comparisons on numerical results to existing BBDF methods demonstrate a comparable performance of both fixed and adaptive step formulas in terms of accuracy. The ρ DIBBDF algorithms are written in C programming language. All the ap- proximate solutions of the standard problems and application systems of stiff ODEs generated by ρ DIBBDF agrees well with the exact solutions and approximate solutions computed by Matlab stiff solvers. All developed methods with ρ = 0.75 have shown to perform the computational work in a lesser time when compared to the existing BBDF methods of the corresponding order. In conclusion, the proposed methods have shown the suitability and reliability to solve linear and non-linear systems in different level of stiffness with comparison to the existing BBDF methods and Matlab stiff solvers. Thus, the new methods developed can be included as viable alternatives for solving first and second order stiff ODEs.

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