Numerical Algorithms Based On Block Multistep Methods For Solutions Of Stiff Ordinary Differential Equations

Numerical solutions for ordinary differential equations (ODEs) are very important inscientific computation, as they are widely used to model real world problems. These modelsoften result to ordinary differential equations when using Mathematical modeling techniquesin simulating the behavior of physical, chemical and biological systems. In order to study thebehavior predicted by a model, the resulting equations must be solved. Since analyticalmethods are not powerful enough to solve some of these models, numerical approach is thusemployed to obtain the approximate solutions to the resulting ordinary differential equations.Therefore, the ability to solve these equations numerically is very important.This thesis presents numerical Algorithm based on linear multistep methods for stiffproblems. These algorithms are formulated so as to be used as a single block method. Threemain class of block implicit linear multistep methods with continuous coefficients werederived via collocation and interpolation approach. The continuous coefficients allow areproduction of additional methods. The advantage of this approach, is that both the mainmethods with the additional methods can be combined and implement them as a single blockmethod without the use of predictors or starting values and thus, making them to be selfstarting with less computational effort. They preserve the properties of the linear multistepmethods but with good convergence and stability properties. The methods include the blockAdam’s type methods, block backward differentiation formulae and the block hybrid methods.The implementation of the methods were done using a fixed step size, the methods useonly the initial value given in the problem to generate the solution. At all blocks except thefirst, the function evaluation is already available from the previous block.An extension to second order method for solution of general second order ordinarydifferential equations is also proposed and its properties discussed. Several numericalexperiments are presented to illustrate the efficiency of the proposed methods.