A brief survey on numerical methods for solving singularly perturbed problems

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Abstract

In the present paper, a brief survey on computational techniques for the different classes of singularly perturbed problems is given. This survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000–2009. However some older important relevant papers are also included in this survey. We also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper, see [Appl. Math. Comput. 30 (1989) 223–259, 130 (2002) 457–510, 134 (2003) 371–429] for details. Thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems.

Introduction

Much scientific endeavour is aimed at the relation between causes and their effects. This becomes more intriguing whenever the cause is small and the effect large. The study of this relation in the field of the theory of perturbations in mathematical or physical systems has already a respectable history [678], which can be retraced to the time of Prandtl about a century ago. Despite this long history the subject is still in a state of vigorous development and it is known as the theory of singular perturbations, where the meaning of a “small” perturbation causing a “large” impact is to be made explicitly clear. The solutions of the singular perturbation problems show a non-uniform behavior as the parameter tends to zero. Singular perturbation problems can be distinguished into two classes, viz singular perturbations of cumulative type and singular perturbations of boundary layer type. The class of singular perturbations of cumulative type concerns oscillating systems where the influence of the small parameter becomes observable only after a long time. An asymptotic approximation of the solution of this class of problems may be obtained by the method of stretching the coordinate t. This method was already introduced at the end of the 19th century by Lindstedt and Poincaré in connection with their studies of perturbation problems in celestial mechanics [404], [405], [507]. This so-called multiple scale technique has been elaborated, refined and applied later on by several others. Another method closely related to the multiple scale technique is based on the averaging principle of Krilov, Bogoliubov and Mitropolski [93].

There are several phenomena in physics and engineering which are characterized by a rapid transition of the observable quantity such as for instance occur in shock waves in gas motions, in boundary layer flow along the surface of a body and in edge effects in the deformation of elastic plates. The mathematical models describing these phenomena contain a small parameter ε and the influence of this parameter reveals itself in a sudden change of the dependent variable uε, taking place within a small layer. The solution of the problem of the boundary layer flow and that of the elastic plate is characterized by the fact that the perturbation with ε small has an observable effect only in the neighborhood of the boundary and, therefore, one uses the term “singular perturbations of boundary layer type”. However, it can also be happen that the perturbation is observable in a thin layer not in the neighborhood of some boundary or edge and in this case we have a “singular perturbation of free layer type”. These thin layers are usually referred to as boundary layers in fluid mechanics, edge layers in solid mechanics, skin layers in electrical applications, shock layers in fluid and solid mechanics, and transition points in quantum mechanics. The mathematical generalization is formulated as follows.

Consider the boundary value problem of the formεL2[uε(x)]+L1[uε(x)]=f(x),xΩRn,0<ε1,where L2 and L1 are differential operators of the orders of m and k respectively with m>k0, while uε(x) satisfies the boundary conditionsBl[uε(x)]=φl(x),xΩ,l=0,1,2,The solution u0 of the reduced equation with ε=0 cannot satisfy in general all boundary conditions and it is clear that the solution uε will show a non-uniform behavior for ε0. This non-uniform behavior manifests itself not only in a small layer in the neighbourhood of the boundary or of some part of the boundary, but also a free layer may occur. It is possible to write uε as a composite expression consisting of two asymptotic expansions, one valid outside the layer, usually called the “outer expansion” and the other valid inside the layer, usually called the “inner expansion”. Much attention has been paid to the matching of the two expansions which should lead to the overall solution uε. Van Dyke [677] introduced certain asymptotic matching principles according to which both asymptotic expansions can be matched and Kaplaun [304] and Lagerstrom [346] introduced the hypothesis that there is a common region where both asymptotic expansions are valid and that an intermediate matching can be applied. The matching procedure may be difficult and it may lead to very unattractive calculations, but on the other hand there are also many cases where the matching can be attained in a rather easy intuitive way. Readers interested in the history of singular perturbations of boundary layer type as well as their theoretical treatment are referred to the books by Ardema [34], Bender and Orszag [63], Cole [138], de Jager and Furu [141], Eckhaus [153], [154], Erdelyi [164], Holmes [254], Kaplaun [304], Kevorkian and Cole [315], [316], Lagerstrom [346], Meyer and Parter [440], Nayfeh [475], [476], O’Malley [487], [488], Smith [610], Van Dyke [677], Verhulst [683], [684], Wasow[718], [719], the papers by Friedrichs [200], Van Dyke [678] and Vishik and Lyusternik [687].

Meanwhile the subject has received a broad international interest stimulated from many research activities and there exists nowadays an overwhelming vast quantity of literature. In the present survey contained in this paper, we do not aim at a complete bibliographical survey. The list of references in this paper includes a number of publications on singularly perturbed steady and unsteady linear/non-linear problems of convection–diffusion and reaction–diffusion type. We have tried our best to cover as much information as we have. However, readers will certainly miss some names and important papers, which is completely unintentional.

Section snippets

Singularly perturbed problems

In 1904 a little-known physicist Ludwig Prandtl revolutionized fluid dynamics with his notion that the effects of friction are experienced only very near an object moving through a fluid. In his groundbreaking paper [510], “Fluid Flow in Very Little Friction, at the Third International Mathematics Congress in Heidelberg, he introduced the concept of boundary layer and its importance for drag and streamlining. Prandtl theorized that an effect of friction was to cause the fluid immediately

Some standard singular perturbation models

In this section, we mention some famous singular perturbation models described in the book by Morton [461], which arise in quite distinct engineering and scientific fields. For more details interested readers can see the references cited along with these models.

Numerical approximations of singularly perturbed problems

During the past few decades, the development of special numerical methods with error independent of the singular perturbation parameter is a well-known direction, in the investigation of such singularly perturbed problems. In this paper, we give a brief survey in a chronological order on the numerical treatment of singularly perturbed problems. Herein we consider a wide variety of singularly perturbed problems, such as one and two dimensional, linear, non-linear, semi-linear, quasi-linear,

1968–1984: Initial era

With the opening words of Morton’s book [461] “Accurate modelling of the interaction between convective and diffusive processes is the most ubiquitous and challenging task in the numerical approximation of partial differential equations”, we shall present an overview of the development of numerical methods for singularly perturbed problems during the last 40 years.

Bobisud [76] considered the behavior of the solution uε of linear second order singularly perturbed parabolic equation εauxx-ut-bux-cu

1985–2000: Mid era

In [533], Reinhardt constructed finite element approximations via modifications of classical methods of lines for two types of singularly perturbed, linear partial differential equations, namely time dependent convection–diffusion problems ut=εuxx-(pu)x-qu+w,(x,t)(0,1)×(0,T],u(0,t)=u(1,t)=0,t(0,T],u(x,0)=g(x),x(0,1) and a two-dimensional elliptic equation -εΔu+ux=w in [0,1]×[0,1]. He established a posteriori estimates for the error between the solutions of the finite element methods and the

2001–2009: Present era

In the year 2001, Vulanović [698] analyzed and compared the Bakhvalov and Shishkin discretization meshes by considering a quasi-linear singularly perturbed boundary value problem without turning points. He also generalized and improved the Shishkin meshes. Langdon and Graham [347] considered boundary integral methods applied to boundary value problems for the positive definite Helmholtz-type problem -ΔU+α2U=0 in a bounded or unbounded domain, with the parameter α real and possibly large.

Conclusion and further directions

This survey paper gives a comprehensive review of state-of-the art numerical techniques used in the solutions of various classes of singularly perturbed problems. We included a large variety of singularly perturbed problems, such as one-dimensional, multi-dimensional, linear, non-linear, semi-linear, quasi-linear, single parameter, multi parameter, and turning point problems in this survey paper. The numerical techniques used by various researchers included in this survey can be classified as

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