Justin B. Munyakazi

On Richardson extrapolation for fitted operator finite difference methods

Abstract Recently, there has been a great interest towards the higher order 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. However, as we see in this article, such techniques do not perform equally well on all type of methods. To investigate this, we consider two fitted operator finite difference methods (FOFDMs) developed by Patidar [K.C. Patidar, High order fitted operator numerical method for self-adjoint singular perturbation problems Appl. Math. Comput. 171(1) (2005) 547–566] and Lubuma and Patidar [J. Lubuma, K.C. Patidar, Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems, J. Comput. Appl. Math. 191 (2006) 228–238], referred to as FOFDM-I and FOFDM-II, respectively. The FOFDM-I is fourth and second order accurate for moderate and smaller values of e, respectively. Unfortunately, Richardson extrapolation does not improve the order of this method. The FOFDM-II is second order uniformly convergent and we show that its order can be improved up to four by using Richardson extrapolation. Both the methods are analyzed for convergence and comparative numerical results supporting theoretical estimates are provided.

A note on the exact discretization for a Cauchy–Euler equation: application to the Black–Scholes equation

We construct the exact finite difference representation for a second-order, linear, Cauchy–Euler ordinary differential equation. This result is then used to construct new non-standard finite difference schemes for the Black–Scholes partial differential equation.

Novel fitted operator finite difference methods for singularly perturbed elliptic convection–diffusion problems in two dimensions

We consider a class of singularly perturbed elliptic problems posed on a unit square. These problems are solved by using fitted mesh methods by many researchers but no attempts are made to solve them using fitted operator methods, except our recent work on reaction–diffusion problems [J.B. Munyakazi and K.C. Patidar, Higher order numerical methods for singularly perturbed elliptic problems, Neural Parallel Sci. Comput. 18(1) (2010), pp. 75–88]. In this paper, we design two fitted operator finite difference methods (FOFDMs) for singularly perturbed convection–diffusion problems which possess solutions with exponential and parabolic boundary layers, respectively. We observe that both of these FOFDMs are ϵ-uniformly convergent. This fact contradicts the claim about singularly perturbed convection–diffusion problems [Miller et al. Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996] that ‘when parabolic boundary layers are present, …, it is not possible to design an ϵ-uniform FOFDM if the mesh is restricted to being a uniform mesh’. We confirm our theoretical findings through computational investigations and also found that we obtain better results than those of Linß and Stynes [Appl. Numer. Math. 31 (1999), pp. 255–270].

PERFORMANCE OF RICHARDSON EXTRAPOLATION ON SOME NUMERICAL METHODS FOR A SINGULARLY PERTURBED TURNING POINT PROBLEM WHOSE SOLUTION HAS BOUNDARY LAYERS

Investigation of the numerical solution of singularly perturb- ed turning point problems dates back to late 1970s. However, due to the presence of layers, not many high order schemes could be developed to solve such problems. On the other hand, one could think of applying the convergence acceleration technique to improve the performance of existing numerical methods. However, that itself posed some challenges. To this end, we design and analyze a novel fitted operator finite difference method (FOFDM) to solve this type of problems. Then we develop a fitted mesh finite difference method (FMFDM). Our detailed convergence analysis shows that this FMFDM is robust with respect to the singular perturbation parameter. Then we investigate the effect of Richardson extrapolation on both of these methods. We observe that, the accuracy is improved in both cases whereas the rate of convergence depends on the particular scheme being used.

A new fitted operator finite difference method to solve systems of evolutionary reaction-diffusion equations

Abstract In recent years, fitted operator finite difference methods (FOFDMs) have been developed for numerous types of singularly perturbed ordinary differential equations. The construction of most of these methods differed though the final outcome remained similar. The most crucial aspect was how the difference operator was designed to approximate the differential operator in question. Very often the approaches for constructing these operators had limited scope in the sense that it was difficult to extend them to solve even simple one-dimensional singularly perturbed partial differential equations. However, in some of our most recent work, we have successfully designed a class of FOFDMs and extended them to solve singularly perturbed time-dependent partial differential equations. In this paper, we design and analyze a robust FOFDM to solve a system of coupled singularly perturbed parabolic reaction-diffusion equations. We use the backward Euler method for the semi-discretization in time. An FOFDM is then developed to solve the resulting set of boundary value problems. The proposed method is analyzed for convergence. Our method is uniformly convergent with order one and two, respectively, in time and space, with respect to the perturbation parameters. Some numerical experiments supporting the theoretical investigations are also presented.