Srinivasan Natesan

A Parallel Boundary Value Technique for Singularly Perturbed Two-Point Boundary Value Problems

A class of singularly perturbed two-point boundary-value problems (BVPs) for second-order ordinary differential equations (DEs) is considered here. In order to obtain numerical solution to these problems, an iterative non-overlapping domain decomposition method is suggested. The BVPs are independent in each subdomain and one can use parallel computers to solve these BVPs. One of the characteristics of the method is that the number of processors available is a free parameter of the method. Practical experiments on a Silicon Graphics Origin 200, with 4 MIPS R10000 processors have been performed, showing the reliability and performance of the proposed parallel schemes. Error estimates for the solution and numerical examples are provided.

A numerical algorithm for singular perturbation problems exhibiting weak boundary layers

Abstract A class of singularly perturbed two-point boundary-value problems (BVP) for second-order ordinary differential equations is considered here. To avoid the numerical difficulties in the solution to these problems, we divide the domain into two subdomains. The first BVP is a layer domain problem and the second BVP is a regular domain problem. Error estimates are derived for the numerical solution. Numerical examples are provided in support of the proposed method.

Parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers

This article presents a numerical method to solve singularly perturbed turning point problems exhibiting two exponential boundary layers. Classical finite-difference schemes do not yield parameter uniform convergent results on a uniform mesh, in general (Robust Computational Techniques for Boundary Layers, Chapman & Hall, London, CRC Press, Boca Raton, FL, 2000). In order to overcome this difficulty, we propose an appropriate piecewise uniform (Shishkin) mesh and apply the classical finite-difference schemes on this mesh. Error estimates are derived by decomposing the solution into smooth and singular components. The present method is layer resolving as well as parameter uniform convergent. Numerical examples are presented to show the applicability and efficiency of the method.

A computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers

Singularly perturbed turning point problems (TPPs) for second order ordinary differential equations (DEs) exhibiting twin boundary layers are considered. In order to obtain numerical solution of these problems a computational method is suggested in which exponentially fitted difference schemes are combined with classical numerical methods. In this method, the given internal (domain of definition of differential equation) is divided into four subintervals and we solve the differential equation in each interval and combine the solution. The implementation of the method on parallel architectures is mentioned. Error estimates are derived and numerical examples are given.

Initial-value technique for singularly-perturbed turning-point problems exhibiting twin boundary layers

The initial-value technique that was originally developed for solving singularly-perturbed nonturning-point problems (Ref. 1) is used here to solve singularly-perturbed turning-point problems exhibiting twin boundary layers. In this method, the required approximate solution is obtained by combining solutions of the reduced problem, an initial-value problem, and a terminal-value problem. Error estimates for approximate solutions are obtained. The implementation of the method on parallel architectures is discussed. Numerical examples are presented to illustrate the present technique.

A “Booster method” for singular perturbation problems arising in chemical reactor theory

A numerical method for singularly perturbed two-point boundary-value problems (BVPs) for second-order ordinary differential equations (ODEs) arising in chemical reactor theory is proposed. In this method, an asymptotic approximation is incorporated into a suitable finite difference scheme to improve the numerical solution. Uniform error estimates are derived when implemented in known difference schemes. Numerical results are presented in support of this claim.

Initial-value technique for singularly perturbed boundary-value problems for second-order ordinary differential equations arising in chemical reactor theory

An initial-value technique is presented for solving singularly perturbed two-point boundary-value problems for linear and semilinear second-order ordinary differential equations arising in chemical reactor theory. In this technique, the required approximate solution is obtained by combining solutions of two terminal-value problems and one initial-value problem which are obtained from the original boundary-value problem through asymptotic expansion procedures. Error estimates for approximate solutions are obtained. Numerical examples are presented to illustrate the present technique.

Richardson extrapolation technique for singularly perturbed parabolic convection–diffusion problems

This paper deals with the study of a post-processing technique for one-dimensional singularly perturbed parabolic convection–diffusion problems exhibiting a regular boundary layer. For discretizing the time derivative, we use the classical backward-Euler method and for the spatial discretization the simple upwind scheme is used on a piecewise-uniform Shishkin mesh. We show that the use of Richardson extrapolation technique improves the ε-uniform accuracy of simple upwinding in the discrete supremum norm from O (N−1 ln N + Δt) to O (N−2 ln2 N + Δt2), where N is the number of mesh-intervals in the spatial direction and Δt is the step size in the temporal direction. The theoretical result is also verified computationally by applying the proposed technique on two test examples.

Numerical methods for elliptic partial differential equations with rapidly oscillating coefficients

This paper presents two methods for the numerical solution of the classical homogenization problem of elliptic operators with periodically oscillating coefficients. The numerical solution of such problems is difficult because of the presence of rapidly oscillating coefficients. The first method based on the classical one which consists of the homogenized solution, the first- and second-order correctors, whereas the second one is based on the Bloch wave approach. Further, for the calculation of the homogenized coefficients and some auxiliary functions involved in this method, we applied both methods and compared their accuracies. The Bloch approximation consists in determining an oscillating integral, numerically. The Bloch method provides a better approximation to the exact solution than the classical first-order corrector term in the smooth coefficients case. Moreover, we provided Taylor approximations for the Bloch approximation function and implemented it numerically. In order to show the efficiency of these methods, exhaustive numerical examples in both one and two-dimensional cases are presented.

FITTED MESH METHOD FOR SINGULARLY PERTURBED REACTION-CONVECTION-DIFFUSION PROBLEMS WITH BOUNDARY AND INTERIOR LAYERS

A robust numerical method for a singularly perturbed secondorder ordinary differential equation having two parameters with a discontinuous source term is presented in this article. Theoretical bounds are derived for the derivatives of the solution and its smooth and singular components. An appropriate piecewise uniform mesh is constructed, and classical upwind finite difference schemes are used on this mesh to obtain the discrete system of equations. Parameter-uniform error bounds for the numerical approximations are established. Numerical results are provided to illustrate the convergence of the numerical approximations.