A brief survey on numerical methods for solving singularly perturbed 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 and the influence of this parameter reveals itself in a sudden change of the dependent variable , 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 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 formwhere and are differential operators of the orders of m and k respectively with , while satisfies the boundary conditionsThe solution of the reduced equation with cannot satisfy in general all boundary conditions and it is clear that the solution will show a non-uniform behavior for . 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 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 . 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 of linear second order singularly perturbed parabolic equation
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 and a two-dimensional elliptic equation in . 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 in a bounded or unbounded domain, with the parameter 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
References (736)
Asymptotic behaviour of solutions near a turning point: the example of the Brusselator equation
J. Differ. Equat.
(2006)The convergence of a finite difference method on layer-adapted mesh for a singularly perturbed system
Appl. Math. Comput.
(2005)- et al.
A note on a parameterized singular perturbation problem
Appl. Math. Comput.
(2005) - et al.
Uniform numerical method for singularly perturbed delay differential equations
Comput. Math. Appl.
(2007) - et al.
Uniform difference method for singularly perturbed volterra integro-differential equations
Appl. Math. Comput.
(2006) - et al.
Singularly perturbed differential equations with discontinuous coefficients and concentrated factors
Appl. Math. Comput.
(2004) - et al.
A note on iterative methods for solving singularly perturbed problems using non-monotone methods on Shishkin meshes
Comput. Methods Appl. Mech. Eng.
(2003) - et al.
Numerical solution of a convection diffusion problem with Robin boundary conditions
J. Comput. Appl. Math.
(2003) - et al.
A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations
J. Comput. Appl. Math.
(2007) - et al.
Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem
Appl. Numer. Math.
(1998)
A multiscale a posteriori error estimate
Comput. Methods Appl. Mech. Eng.
An adaptive stabilized finite element scheme for the advection–reaction–diffusion equation
Appl. Numer. Math.
Error estimators for advection–reaction–diffusion equations based on the solution of local problems
J. Comput. Appl. Math.
Coupling stabilized finite element methods with finite difference time integration for advection–diffusion–reaction problems
Comput. Methods Appl. Mech. Eng.
A numerical algorithm for some singularly perturbed boundary value problems
J. Comput. Appl. Math.
A combined method of local Green’s functions and central difference method for singularly perturbed convection–diffusion problems
J. Comput. Appl. Math.
Quasioptimal finite element approximations of first order hyperbolic and of convection-dominated convection–diffusion equations
North-Holland Math. Stud.
A spline method for second-order singularly perturbed boundary-value problems
J. Comput. Appl. Math.
An error analysis for the finite element method applied to convection diffusion problems
Comput. Methods Appl. Mech. Eng.
Approximate symmetrization and Petrov–Galerkin methods for diffusion–convection problems
Comput. Methods Appl. Mech. Eng.
Spline based computational technique for linear singularly perturbed boundary value problems
Appl. Math. Comput.
On a uniformly accurate finite difference approximation of a singularly perturbed reaction–diffusion problem using grid equidistribution
J. Comput. Appl. Math.
Uniformly convergent high order finite element solutions of a singularly perturbed reaction–diffusion equation using mesh equidistribution
Appl. Numer. Math.
A parameter robust numerical method for a system of two singularly perturbed convection–diffusion equations
Appl. Numer. Math.
New substructuring domain decomposition methods for advection–diffusion equations
J. Comput. Appl. Math.
Negative norm stabilization of convection–diffusion problems
Appl. Math. Lett.
Downwind numbering: robust multigrid for convection–diffusion problems
Appl. Numer. Math.
Experiments with a local adaptive grid h-refinement for the finite-difference solution of BVPs in singularly perturbed second-order ODEs
Appl. Math. Comput.
Difference methods for solving convection–diffusion equations
Comput. Math. Appl.
The projection method for singularly perturbed boundary-value problems
USSR Comput. Math. Math. Phys.
Stability of the SUPG finite element method for transient advection–diffusion problems
Comput. Methods Appl. Mech. Eng.
Domain decomposition for a singularly perturbed parabolic problem with a convection-dominated term
J. Comput. Appl. Math.
Schwarz alternating algorithms for a convection–diffusion problem
Appl. Math. Comput.
A block monotone domain decomposition algorithm for a semilinear convection–diffusion problem
J. Comput. Appl. Math.
Monotone Schwarz iterates for a semilinear parabolic convection–diffusion problem
J. Comput. Appl. Math.
A uniformly convergent method for a singularly perturbed semilinear reaction–diffusion problem with discontinuous data
Appl. Math. Comput.
The solution of a semilinear evolutionary convection–diffusion problem by a monotone domain decomposition algorithm
Appl. Math. Comput.
Uniform convergence of a weighted average scheme for a nonlinear reaction–diffusion problem
J. Comput. Appl. Math.
Uniform numerical methods on arbitrary meshes for singularly perturbed problems with discontinuous data
Appl. Math. Comput.
Iterative domain decomposition algorithms for a convection–diffusion problem
Comput. Math. Appl.
On an uniform multidomain decomposition method applied to a singularly perturbed problem with regular boundary layers
J. Comput. Appl. Math.
Numerical solution of some quasilinear singularly perturbed heat-conduction equations on nonuniform grids
USSR Comput. Math. Math. Phys.
Uniformly convergent finite volume difference scheme for 2D convection-dominated problem with discontinuous coefficients
Appl. Math. Comput.
Numerical solution of a two-dimensional singularly perturbed reaction–diffusion problem with discontinuous coefficients
Appl. Math. Comput.
Numerical solution of a mixed singularly perturbed parabolic–elliptic problem
J. Math. Anal. Appl.
On the choice of a stabilizing subgrid for convection–diffusion problems
Comput. Methods Appl. Mech. Eng.
Stability and error analysis of mixed finite-volume methods for advection dominated problems
Comput. Math. Appl.
A priori estimates for solutions of singular perturbations with a turning point
Stud. Appl. Math.
Difference approximations for singular perturbations of systems of ordinary differential equations
Numer. Math.
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