Uniform and optimal schemes for stiff initial-value problems

AbstractWe formulate a class of difference schemes for stiff initial-value problems, with a small parameter ϵ multiplying the first derivative. We derive necessary conditions for uniform convergence with respect to the small parameter ϵ, that is the solution of the difference scheme uih satisfies |uih−u(xi)| ⪕ Ch, where C is independent of h and ϵ. We also derive sufficient conditions for uniform convergence and show that a subclass of schemes is also optimal in the sense that |uih−u(xi)| ⪕ C min (h, ϵ). Finally, we show that this class contains higher-order schemes

An Initial Value Technique using Exponentially Fitted Non Standard Finite Difference Method for Singularly Perturbed Differential-Difference Equations

In this paper, an exponentially fitted non standard finite difference method is proposed to solve singularly perturbed differential-difference equations with boundary layer on left and right sides of the interval. In this method, the original second order differential difference equation is replaced by an asymptotically equivalent singularly perturbed problem and in turn the problem is replaced by an asymptotically equivalent first order problem. This initial value problem is solve by using exponential fitting with non standard finite differences. To validate the applicability of the method, several model examples have been solved by taking different values for the delay parameter δ , advanced parameter η and the perturbation parameter ε . Comparison of the results is shown to justify the method. The effect of the small shifts on the boundary layer solutions has been investigated and presented in figures. The convergence of the scheme has also been investigated

Numerical Treatment of Non-Linear singular pertubation problems

Magister Scientiae - MScThis thesis deals with the design and implementation of some novel numerical methods for non-linear singular pertubations problems (NSPPs). It provide a survey of asymptotic and numerical methods for some NSPPs in the past decade. By considering two test problems, rigorous asymptotic analysis is carried out. Based on this analysis, suitable numerical methods are designed, analyzed and implemented in order to have some relevant results of physical importance. Since the asymptotic analysis provides only qualitative information, the focus is more on the numerical analysis of the problem which provides the quantitative information.South Afric

Higher order numerical methods for singular perturbation problems

Philosophiae Doctor - PhDIn recent years, there has been a great interest towards the higher order numerical methods for singularly perturbed problems. As compared to their lower order counterparts, they provide better accuracy with fewer mesh points. Construction and/or implementation of direct higher order methods is usually very complicated. Thus a natural choice is to use some convergence acceleration techniques, e.g., Richardson extrapolation, defect correction, etc. In this thesis, we will consider various classes of problems described by singularly perturbed ordinary and partial differential equations. For these problems, we design some novel numerical methods and attempt to increase their accuracy as well as the order of convergence. We also do the same for existing numerical methods in some instances. We find that, even though the Richardson extrapolation technique always improves the accuracy, it does not perform equally well when applied to different methods for certain classes of problems. Moreover, while in some cases it improves the order of convergence, in other cases it does not. These issues are discussed in this thesis for linear and nonlinear singularly perturbed ODEs as well as PDEs. Extrapolation techniques are analyzed thoroughly in all the cases, whereas the limitations of the defect correction approach for certain problems is indicated at the end of the thesis.South Afric