The " { uniform convergence of the discretederivativesfor singularly perturbed problems

نویسندگان

  • P. A. Farrell
  • A. F. Hegarty
  • J. J. H. Miller
چکیده

The derivatives of the solution of singularly perturbed diierential equations become unbounded as the singular perturbation parameter " tends to zero. Therefore to approximate such derivatives, it is required to scale the derivatives in such a way that they are of order one for all values of the perturbation parameter. In practice , derivatives are related to the ux or drag and, hence, it is desirable to have "{ uniform approximations to the scaled derivatives. In this paper, singularly perturbed convection{diiusion problems are considered. The use of standard scaled discrete derivatives to approximate the scaled continuous derivatives of the solution of singularly perturbed problems is examined. Standard scaled discrete derivatives generated from exact numerical methods on a uniform mesh are shown to be not "{uniformly convergent. On the other hand, standard scaled discrete derivatives computed from a numerical method based on an appropriately tted piecewise{uniform mesh are shown to be "{uniformly convergent. Numerical results are presented and discussed to illustrate the signiicance of these theoretical results.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

An efficient numerical method for singularly perturbed second order ordinary differential equation

In this paper an exponentially fitted finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer. A fitting factor is introduced and the model equation is discretized by a finite difference scheme on an uniform mesh. Thomas algorithm is used to solve the tri-diagonal system. The stability of the algorithm is investigated. It ...

متن کامل

A Parameter Uniform Numerical Scheme for Singularly Perturbed Differential-difference Equations with Mixed Shifts

In this paper, we consider a second-order singularly perturbed differential-difference equations with mixed delay and advance parameters. At first, we approximate the model problem by an upwind finite difference scheme on a Shishkin mesh. We know that the upwind scheme is stable and its solution is oscillation free, but it gives lower order of accuracy. So, to increase the convergence, we propo...

متن کامل

Numerical method for a system of second order singularly perturbed turning point problems

In this paper, a parameter uniform numerical method based on Shishkin mesh is suggested to solve a system of second order singularly perturbed differential equations with a turning point exhibiting boundary layers. It is assumed that both equations have a turning point at the same point. An appropriate piecewise uniform mesh is considered and a classical finite difference scheme is applied on t...

متن کامل

A hybrid method for singularly perturbed delay boundary value problems exhibiting a right boundary layer

The aim of this paper is to present a numerical method for singularly perturbed convection-diffusion problems with a delay. The method is a combination of the asymptotic expansion technique and the reproducing kernel method (RKM). First an asymptotic expansion for the solution of the given singularly perturbed delayed boundary value problem is constructed. Then the reduced regular delayed diffe...

متن کامل

Numerical method for singularly perturbed fourth order ordinary differential equations of convection-diffusion type

In this paper, we have proposed a numerical method for singularly perturbed  fourth order ordinary differential equations of convection-diffusion type. The numerical method combines boundary value technique, asymptotic expansion approximation, shooting method and  finite difference method. In order to get a numerical solution for the derivative of the solution, the given interval is divided  in...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 1999