numerical integration of non linear two point boundary value problems using iterated deferred corrections i abstract in several recent papers the idea has been proposed of using implicit runge kutta formulae for the approximate integration of non linear two point boundary value problems these formulae often take a very special form so that computational efficiencies are obtained in certain well defined circumstances in this paper we examine the relative merits of these different classes of formulae and discuss their efficient implementation in a deferred correction framework a theoretical comparison of these formulae is important firstly because they call for different computational tasks to be performed and secondly because they are not always applicable to exactly the same classes of problems the most efficient of these formulae are then compared with methods based on finite differences and with methods based on spline collocation for some simple smooth test functions on the basis of operation counts and numerical experimentation it is concluded that runge kutta methods can be generally competitive with the other methods considered

third order variable mesh cubic spline methods for singularly perturbed boundary value problems abstract third order variable mesh methods based on cubic spline approximation for nonlinear singularly perturbed boundary value problems of the form ey f x y y a y b are presented the convergence analysis is given and the method is shown to have third order convergence several tet examples are solved to demonstrate the efficiency of the method

solving singularly perturbed boundary value problems by spline in tension the difference scheme via spline in tension for the problem y p x y f x p x s 0 y 0 0 y 1 1 is derived the error of the form o h min h is obtained when p x p const the corresponding spline in tension achieves the second order of the global uniform convergence

reply to the comment of e kahya on an improvement on fibonacci search method in optimization theory by murat subasi necmettin yildirim bunyamin yildiz an improvement on fibonacci search method in optimization theory applied mathematics and computation 147 2004 893 901 abstract this paper includes a reply to the comment suggested by kahya emin kahya comment on titled an improvement on fibonacci search method in optimization theory kahya claims that he proposes some corrections for some deficiencies of the paper murat subasi necmettin yildirim bunyamin yildiz an improvement on fibonacci search method in optimization theory applied mathematics and computation 147 2004 893 901 here we prove that in kahya s suggestions there is no novelty and so they are not contributional

an algebraic approximation for the matrix exponential in singularly perturbed boundary value problems a symmetric difference scheme for linear stiff or singularly perturbed boundary value problems of first order with constant coefficients is constructed being based on a stability function containing a matrix square root its essential feature is the unconditional stability in the absence of purely imaginary eigenvalues of the coefficient matrix local damping of errors uniform stability and uniform second order convergence are proved the computation of the specific matrix square root by a well known stable variant of newton s method is discussed a numerical example confirming the results is given

an adaptive uniformly convergent numerical method for a semilinear singular perturbation problem a singularly perturbed semilinear two point boundary value problem is considered without any restriction on its turning points a difference scheme is presented for solving this problem on an arbitrary locally quasi uniform mesh it is shown that the solution of the scheme is first order accurate uniformly in the perturbation parameter in a discrete l 1 norm numerical results are presented for an adaptive method based on the scheme

an improvement on fibonacci search method in optimization theory among the other elimination methods fibonacci search method is regarded as the best one to find the optimal point for single valued functions in the present study employing lucas numbers instead of fibonacci numbers we have made partial improvements on location of the intervals that contain optimal point in the fibonacci search algorithm for that purpose using two well known test functions in optimization theory a computer program was developed in maple to examine this idea it was seen that the improved method is giving better results than the previous method in the sense that converging the optimal point more rapidly

on some singularly perturbed problems for viscous stratified fluids abstract the singularly perturbed boundary value problem which describes the motion of a stratified viscous fluid is obtained and an asymptotic expansion of the solution for vanishing viscosity is developed the two cases considered are when the reduced equation has i a unique solution and ii a family of solutions the latter situation is an example of a singular singular perturbation problem proof of the uniform asymptotic validity of the constructed expansion is given

on smoothing and order reduction effects for implicit runge kutta formulae abstract it is well known that many important classes of runge kutta methods suffer an order reduction phenomenon when applied to certain classes of stiff problems in particular the s stage gauss methods with stage order s and order of consistency 2s behave like methods of order s when applied to the class of singularly perturbed problems in this paper we will show that the process of smoothing can ameliorate this effect when dealing with initial value problems by first studying the effect of smoothing on the standard prothero robinson problem and then by extending the analysis to the general class of singularly perturbed problems

a uniformly accurate finite element method for a singularly perturbed one dimensional reaction diffusion application d une methode a elements finis avec des elements de base exponentiels a un probleme autoadjoint a perturbation singuliere et a deux points limites generation d un schema aux differences tridiagonal uniformement precis du second ordre

a riccati transformation method for solving linear bvps ii computational aspects in part i of this paper siamj numer anal 25 1988 pp 1055 1073 a riccati transformation method for solving linear boundary value problems for ordinary differential equations is presented here the implementation aspects of this method are discussed in particular a double sweep integration process which includes a strategy for solving a matrix riccati differential equation and forming a continuous representation of its solution is described the efficacy of the method is demonstrated on a number of singular perturbation problems particular attention is paid to the choice of scaling for the variables and its effect on the solution method the computational experience while not based upon a refined implementation indicates that the riccati method may well prove to be competitive with the other general methods

stability of discrete delay systems boundary criteria this paper deals with the stability of linear discrete delay systems two stability criteria that are described by evaluating a corresponding harmonic function on the boundary of a torus region are presented testing these criteria can be proved to be much easier than those reported in the literature

direct scheme in optimal control problems with fast and slow motions abstract the nonlinear optimal control problems described by singularly perturbed models are studied in order to obtain the ways of decomposition and to construct the suboptimal control sequences a new scheme the so called direct scheme applying the vasil eva boundary layer functions method in control theory is introduced it consists in directly expanding the problem s conditions into postulated asymptotic series and then in successively solving lower dimensional approximate decomposition problems of optimal control by means of the direct scheme a new property of an asymptotic expansion the relaxation is obtained i e the decrease of the performance index with each new control approximation illustrating examples are given

on solutions of some unsteady flows over a continuous moving porous flat surface abstract a singular perturbation solution for hydromagnetic boundary layer flow over a continuously moving surface with large suction is obtained and compared with its exact solution also the solutions are obtained for finite and large values of t time and for various asymptotic cases flow of this type represents a new class of boundary layer problems with solutions substantially different from those for boundary layer flow on a flat surface of finite length surprisingly the inner solution of the problem is valid even in the outer region and the effect of magnetic field on the flow field is almost negligible furthermore it is shown that a straightforward perturbation expansion valid for all t time and y space coordinate for small suction is not possible because of the infinite condition on y

approximate method for the numerical solution of singular perturbation problems we present an approximate method for the numerical solution of linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval it is motivated by the asymptotic behavior of singular perturbation problems the original problem is divided into inner and outer region problems the reduced problem is solved to obtain the terminal boundary condition then a new inner region problem is created and solved as a two point boundary value problem in turn the outer region problem is also modified and the resulting problem is efficiently treated by employing the trapezoidal formula coupled with discrete invariant imbedding algorithm the proposed method is iterative on the terminal point some numerical experiments have been included to demonstrate its applicability

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods

higher order uniformly convergent methods for singular perturbation problems abstract until now the best way to construct high order uniformly convergent schemes for singular perturbation problems was to apply exponentially fitted compact difference schemes we propose the following approach firstly solve an auxiliary problem with piecewise or nearly piecewise constant coefficients secondly improve the approximation iteratively using the defect correction idea and piecewise polynomial approximations of higher order an important advantage of our approach lies in the fact that it is possible to start from a classical or weak formulation of a boundary value problem therefore the approach is useful for singular perturbations related to ordinary as well as partial differential equations

a computational method for solving singular perturbation problems using exponentially fitted finite difference schemes abstract a computational method is presented for solving linear singularly perturbed two point boundary value problems with a boundary layer on the left end of the interval the original problem is divided into inner and outer region problems the zeroth order asymptotic expansion is used to obtain the terminal boundary condition then a new inner region problem irp is created and solved as a two point boundary value problem tpbvp in turn the outer region problem orp is also solved as a tpbvp both these problems tpbvps are efficiently solved by employing uniform and optimal exponentially fitted finite difference schemes

a class of simple exponential b splines and their application to numerical solution to singular perturbation problems we shall consider an application of simple exponential splines to the numerical solution of singular perturbation problem the computational effort involved in our collocation method is less than that required for the other methods of exponential type

a numerical scheme to approximate the solution of a singularly perturbed nonlinear differential equation abstract in this paper an analytic difference scheme to integrate the singularly perturbed scalar problem 1 1 is given the algorithm is based on the exact integration of a locally linearized problem on a special nonuniform mesh exhibiting uniform convergence in for any x the scheme is exact when f a bx cy where a b c are constants finally numerical results are given

higher order chebyshev basis functions for two point boundary value problems cubic basis functions in one dimension for the solution of two point boundary value problems are constructed based on the zeros of chebyshev polynomials of the first kind a general formula is derived for the construction of polynomial basis functions of degree r where 1 r a galerkin finite element method using the constructed basis functions for the cases r 1 2 and 3 is successfully applied to three different types of problem including a singular perturbation problem

a review on singularly perturbed differential equations with turning points and interior layers singular perturbation problems with turning points arise as mathematical models for various physical phenomena the problem with interior turning point represent one dimensional version of stationary convection diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin boundary turning point problems on the other hand arise in geophysics and in modeling thermal boundary layers in laminar flow in this paper we review some existing literature on asymptotic and numerical analysis of singularly perturbed turning point and interior layer problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye towards the goal of developing general methods to solve such problems since major work in this area started after 1970 so this paper limits its coverage to the work done by numerous researchers between 1970 and 2011

asymptotic expansion for the initial value problem of the sunflower equation abstract in this paper we consider the initial value problem ivp for the sunflower equation ex t ax t b sin x t e 0 with x t t t e 0 x 0 y0 we prove that the solution of the above mentioned ivp has an asymptotic expansion uniformly valid on e for e sufficiently small

a finite element method for a singularly perturbed boundary value problem we examine the problem eu a x u b x u f x for 0 0 b x 2 4e 0 a b andf inc 2 0 1 e in 0 1 u 0 andu 1 given using finite elements and a discretized green s function we show that the el mistikawy and werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem i e the nodal errors are bounded bych 2 wherec is independent ofh and e with a natural choice of trial functions uniform first order accuracy is obtained in thel 0 1 norm on choosing piecewise linear trial functions hat functions uniform first order accuracy is obtained in thel 1 0 1 norm

asymptotic and numerical analysis of singular perturbation problems a survey this paper is intended to be a brief survey of the asymptotic and numerical analysis of singular perturbation problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye toward the goal of developing general methods to solve such problems a summary of the results of some recent methods is presented and this leads to conclusions and recommendations about what methods to use on singular perturbation problems finally some areas of current research are indicated a bibliography of about 130 items is provided

optimal order nonnested multigrid methods for solving finite element equations iii on degenerate meshes in this paper we consider several model problems where finite element triangular meshes with arbitrarily small angles high aspect ratios are utilized to deal with anisotropy interfaces or singular perturbations the constant rate independent of the number of unknowns the smallest angle the interface discontinuity the singular perturbation parameter etc convergence of some special nonnested multigrid methods for solving the finite element systems on such degenerate meshes will be proved numerical data are provided to support the analysis in each case

delay independent exponential stability criteria for time varying discrete delay systems derives delay independent exponential stability conditions for linear nonlinear time varying discrete delay systems since these conditions are of delay independence and easily verifiable they may provide handy tools for the stability analysis

a uniformly convergent method for a singularly perturbed semilinear reaction diffusion problem with discontinuous data this paper deals with a uniform in a perturbation parameter convergent difference scheme for solving a nonlinear singularly perturbed two point boundary value problem with discontinuous data of a reaction diffusion type an error analysis is based on locally exact schemes uniform convergence of the proposed difference scheme on piecewise uniform and log meshes is proven a monotone iterative method which is based on the method of upper and lower solutions is applied to computing the nonlinear difference scheme numerical experiments are presented

a singularly perturbed boundary value problem modelling a semiconductor device this paper is concerned with the static one dimensional modelling of a semiconductor device namely the pn junction when a bias is applied the governing equations are the well known equations describing carrier transport in a semiconductor which consist of a system of ordinary differential equations subject to boundary conditions imposed at the contacts because of the different orders of magnitude of the solution components at the boundaries we scale the components individually and obtain a singular perturbation problem we analyse the equilibrium case zero bias applied and set up approximate models posed as singularly perturbed second order equations by neglecting the hole and electron current densities this makes sense for small forward bias and for moderate reverse bias for the full problem we prove an a priori estimate on the number of electron hole carrier pairs and derive asymptotic expansions as the perturbation parameter tends to zero by setting up the reduced system and the boundary

uniform high order difference schemes for a singularly perturbed two point boundary value problem on construit une famille de schemas aux differences finies uniformement precis pour le probleme du type eu a x u b x u f x on donne une analyse complete de l erreur de discretisation basee sur les resultats de stabilite on presente des essais numeriques

application of global methods in parallel shooting a new divide and conquer method is proposed it employs a coarse grid discretization like multiple shooting does but solves the local problems in a bvp way i e by some global method a sophisticated error control is developed to combine local and global convergence of the newton updating an implementation based on using the collocation code colnew is briefly discussed and a number of examples are given to illustrate the success of the method in particular for singular perturbation problems

l 1 and l uniform convergence of a difference scheme for a semilinear singular perturbation problem a nonlinear difference scheme is given for solving a semilinear singularly perturbed two point boundary value problem without any restriction on turning points the solution of the scheme is shown to be first order accurate in the discretel1 norm uniformly in the perturbation parameter when turning points are excluded the scheme is first order accurate in the discretel norm uniformly in the perturbation parameter

the method of inner boundary condition a new approach for solving singular perturbation problems abstract the object of this paper is to present a new approach based on the method of inner boundary condition for solving singular perturbation problems the original problem is partitioned into inner and outer region differential equation systems asymptotic expansion is used to obtain the terminal boundary condition using an appropriate transformation a new inner region problem is obtained and solved as a two point boundary value problem the derivative boundary condition at the terminal point is then derived from the solution of the inner region problem using this condition the outer region problem is efficiently solved by employing the classical finite difference scheme the proposed method is iterative on the terminal point some numerical examples have been solved to demonstrate the efficiency of the method

first asymptotic approximations to a solution of singularly perturbed optimal control problem with intersecting solutions of degenerate problem using the direct scheme method for constructing asymptotic solution of optimal control problems with small parameters consisting of immediate substituting a postulated asymptotic expansion of a solution into the problem condition and receiving problems for finding asymptotic terms we construct formally the zero order asymptotic approximation to an optimal control and the first order approximation to an optimal trajectory of a singularly perturbed optimal control problem with a weakly controllable state equation a cheap control a not uniquely solvable degenerate state equation fixed endpoints and with intersecting trajectories of the degenerate state equation corresponding to slow trajectories of the perturbed problem together with boundary layer functions in the vicinities of both ends of the considered interval the constructed asymptotics contains inner boundary layer functions an illustrative example is given

simple stability criteria for nonlinear time varying discrete systems abstract we present two sufficient conditions for asymptotic stability of nonlinear time varying discrete systems our main results generalize mori s result for linear discrete time invariant systems to nonlinear time varying systems

on a difference scheme of exponential type for a nonlinear singular perturbation problem a difference scheme of exponential type for solving a nonlinear singular perturbation problem is analysed although this scheme is not of monotone type al 1 convergence result is obtained relations between this scheme and engquist osher scheme are also discussed

an automatic continuation strategy for the solution of singularly perturbed linear two point boundary value problems an automatic continuation alogorithm for the solution of linear singular perturbation problems is developed and incorporated into two codes which implement global methods for solving two point boundary value problems specifically the algorithm will be based upon error estimates formulated in the collocation code colsys and the deferred correction code hagron the benefits of using continuation are clearly demonstrated for both codes for a large class of problems

an analysis of a uniformly convergent finite difference finite element scheme for a model singular perturbation problem on etablit la convergence uniforme de la discretisation d un probleme a perturbation singuliere on utilise quelques resultats de stabilite et un theoreme de representation des solutions

a schwarz alternating procedure for singular perturbation problems we show that the schwarz alternating procedure offers a good algorithm for the numerical solution of singular perturbation problems provided the domain decomposition is properly designed to resolve the boundary and transition layers we give sharp estimates for the optimal position of the domain boundaries and study the convergence rates of the algorithm for various linear second order singular perturbation problems we report on implementation results for a turning point problem and a combustion problem

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

numerical integration method for general singularly perturbed two point boundary value problems in this paper a numerical integration method is presented for solving general singularly perturbed two point boundary value problems the original second order differential equation is replaced by an approximate first order differential equation with a small deviating argument then the trapezoidal formula is used to obtain the three term recurrence relationship the proposed method is iterative on the deviating argument to demonstrate the applicability of the method we have solved several model linear and non linear examples with left end boundary layer or right end boundary layer or an internal layer or two boundary layers and presented the computational results

multigrid applied to singular perturbation problems the solution of the singular perturbation problem email protected u b x u f 0 0 u 0 u 0 u 1 u 1 by a multigrid algorithm is considered theoretical and experimental results for a number of different discretizations are presented the theoretical and observed rates agree with the results developed in an earlier work of kamowitz and parter in addition the rate of convergence of the algorithm when the coarse grid operator is the natural finite difference analogue of the fine grid operator is presented this is in contrast to the case in the previous work where the galerkin choice i h hl hi h h was used for the coarse grid operators

a uniformly accurate finite element method for a singular perturbation problem in conservative form a petrov galerkin finite element method with exponential basis elements is applied to a nonselfadjoint singularly perturbed two point boundary value problem in conservative form it is shown to be uniformly first order accurate in l infty and is uniformly second order accurate at the nodes i e the error constants are independent of the mesh size and the perturbation parameter

analysis of difference schemes for a stationary shock problem we consider the limit function u of the solutions u varepsilon of a singularly perturbed boundary value problem in general u will be discontinuous two difference approximations of the boundary value problem are studied and the discrete values for varepsilon 0 h to 0 are compared with u error estimates of order o h and o h 2 are proved and the sharpness of numerical layers is established

a uniform finite element method for a conservative singularly perturbed problem abstract we examine the problem p x u q x u r x u f x for 0 x p 0 q 0 r 0 p q r and f in c 2 0 1 in 0 1 u 0 and u 1 given existence of a unique solution u and bounds on u and its derivatives are obtained using finite elements on an equidistant mesh of width h we generate a tridiagonal difference scheme which is shown to be uniformly second order accurate for this problem i e the nodal errors are bounded by ch 2 where c is independent of h and with a natural choice of trial functions uniform first order accuracy is obtained in the l 0 1 norm using trial functions which interpolate linearly between the nodal values generated by the difference scheme gives uniform first order accuracy in the l 1 0 1 norm

a brief survey on numerical methods for solving singularly perturbed problems 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

the numerical solution of discrete delay systems in this note we consider the numerical solution of initial value discrete delay systems an interpolation procedure is introduced to compute the numerical solution the convergence and stability of the interpolation procedure for the discrete delay systems is discussed

monotone difference schemes for singular perturbation problems we consider numerical solutions of the boundary value problem varepsilon y f y b x y 0 0 leqq x leqq 1 y 0 a y 1 b varepsilon 0 b y geqq delta 0 with monotone difference schemes on nonuniform grids we prove general convergence results and show that the engquist osher monotone scheme will reproduce essential properties of the true solution for any grid for the inversion of the nonlinear scheme we suggest implicit time relaxation with variable time steps

some improvements of the nonequidistant engquist osher scheme two nonequidistant finite difference schemes for quasilinear singular perturbation problems are given and their properties are discussed and compared one of them corresponds to the equidistant lorenz modification of the engquist osher scheme the other one switches in dependence on the cell reynolds number both schemes show quadratic l 1 accuracy but the second one gives better pointwise results

applications of singular perturbation techniques to control problems this paper discusses typical applications of singular perturbation techniques to control problems in the last fifteen years the first three sections are devoted to the standard model and its time scale stability and controllability properties the next two sections deal with linear quadratic optimal control and one with cheap near singular control then the composite control and trajectory optimization are considered in two sections and stochastic control in one section the last section returns to the problem of modeling this time in the context of large scale systems the bibliography contains more than 250 titles

asymptotic solution of a linear quadratic problem with discontinuous coefficients and cheap control abstract this paper is devoted to a linear quadratic problem with cheap control and discontinuous coefficients using the direct scheme consisting of an immediate substitution of a postulated asymptotic expansion of a solution into the problem condition and finding a family of control problems to define the terms of the asymptotic expansion the asymptotic solution of the considered problem is obtained the proximity of the exact solution to the approximate asymptotic one is estimated for the control trajectory and functional it is shown that the minimized functional does not increase when a higher order asymptotic approximation to the optimal control is used an illustrative example is presented

on dichotomy and well conditioning in bvp we investigate the relationships between the stability bounds of the problem on the one hand and the growth behaviour of the fundamental solution on the other hand it is shown that if these stability bounds are moderate i e if the problem is well conditioned then the homogeneous solution space is dichotomic which means that it can be split into a subspace of nondecreasing and a complementary subspace of nonincreasing modes this is done by carefully examining the green s functions if these exhibit an exponential behaviour then the solution space is also exponentially dichotomic on the other hand we also show that exponential dichotomy implies moderate stability constants i e well conditioning from this it follows that both concepts are more or less equivalent

comment on titled an improvement on fibonacci search method in optimization theory this paper includes a critical review to the paper suggested by subasi et al m subasi n yildirim b yildiz an improvement on fibonacci search method in optimization theory applied mathematics and computation 147 3 2004 893 901 they satisfied an improvement on the classical fibonacci search algorithm using lucas numbers instead of conventional fibonacci numbers however in their algorithm some deficiencies reducing the efficiency of the new method and also incorrect statements have been found

multigrid methods for variational problems this paper develops a very simple but powerful theory for multigrid methods which applies directly to variationally posed operator equations

a singular perturbation analysis of the transient semiconductor device equations this paper is concerned with the asymptotic analysis of the transient semiconductor device equations by appropriately scaling the equations the semiconductor device problem is reformulated as a si

a new method of stabilization for singular perturbation problems with spectral methods a stabilized treatment of singular perturbation problems with spectral methods is presented the stabilization is achieved by introducing one additional collocation point the resulting overdetermined system leads to the normal equations which yield a stabilized system numerical results in one and two dimensions are presented which show the improved condition number of the stabilized spectral system

variable mesh spline approximation method for solving singularly perturbed turning point problems having boundary layer s abstract a numerical method based on cubic spline with nonuniform grid is given for the singularly perturbed two point boundary value problems having turning point the scheme derived in this paper is second order accurate numerical examples are given to support the predicted theory

strong uniform stability and exact discretizations of a model singular perturbation problem and its finite difference approximations this paper is concerned with stability with respect to norms like u e u e u uniformly in e for the operator l eu eu a x u b x u with a x a 0 and compact finite difference discretizations of it strong uniform stability of the continuous problem is established for all e 0 sufficiently small strong stability of the discrete operators is analyzed using the general stability results of niederdrenk and yserentant numer math 41 1983 pp 223 253 as well as from the stand point of exact discretizations examples illustrating some of the anomalous behavior associated with these problems schemes can be compact and uniformly consistent but not uniformly stable for instance are presented

global uniformly convergent schemes for a singularly perturbed boundary value problem using patched base spline functions abstract for a self adjoint second order model singular perturbation problem global uniformly convergent schemes are constructed the basic idea consists in replacing the coefficients by piecewise polynomials and solving the resulting problems exactly

a fast adaptive numerical method for stiff two point boundary value problems we describe a robust adaptive algorithm for the solution of singularly perturbed two point boundary value problems many different phenomena can arise in such problems including boundary layers dense oscillations and complicated or ill conditioned internal transition regions working with an integral equation reformulation of the original differential equation we introduce a method for error analysis which can be used for mesh refinement even when the solution computed on the current mesh is underresolved based on this method we have constructed a black box code for stiff problems which automatically generates an adaptive mesh resolving all features of the solution the solver is direct and of arbitrarily high order accuracy and requires an amount of time proportional to the number of grid points

computational method for a singular perturbation problem via domain decomposition and its parallel implementation abstract in this paper a computation method for the nonlinear problem with a small perturbation parameter via domain decomposition is presented we apply a combination of the two approaches to solving the problem iterative algorithms for domain decomposition and a grid refinement technique on subdomains two iterative algorithms are examined the first one is the schwarz alternating procedure and the second algorithm is highly suitable for parallel computing numerical examples are provided

a study of some multigrid ideas in an effort to understand certain ideas and concepts associated with multigrid iterations we give an in depth study of a particular simple problem we consider a standard finite difference system associated with the two point boundary value problem pu bu qu 0 u 0 u 1 0 the operators i 2 h h i h 2 h are operator based interpolation and projection operators while the smoothers are the damped jacobi iterations with parameter a 0 we determine the exact rates of convergence for the two grid scheme and upper bounds 1 for the multigrid schemes experimental results are discussed

uniform fourth order difference scheme for a singular perturbation problem the numerical solution of a nonlinear singularly perturbed two point boundary value problem is studied the developed method is based on hermitian approximation of the second derivative on special discretization mesh numerical examples which demonstrate the effectiveness of the method are presented

on optimally scaled systems for second order scalar singularity perturbed problems the conditioning of the singularly perturbed scalar dirichlet problem is considered it is shown how this is related to the conditioning of an appropriate associated first order system through this the dichotomy of the solution space a concept that only makes sense in a vectorial setting can be investigated two typical equations are studied in more detail one with possible boundary layers on both sides of the interval and one with an internal layer i e the turning point case the results are applied to obtain estimates of global discretization errors for difference methods several examples illustrate the analysis

numerical solution of singular perturbation problems via deviating arguments we present a new approach to numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the interval the original problem is divided into outer and inner region problems a terminal boundary condition in the implicit form is derived then the outer region problem is solved as a two point boundary value problem tpbvp and an explicit terminal boundary condition is obtained in turn a new inner region problem is obtained and solved as a tpbvp using the explicit terminal boundary condition the proposed method is iterative on the terminal point of the inner region some numerical examples have been solved to demonstrate the applicability of the method

accurate discretization for singular perturbations the one dimensional case this paper develops a discretization method for one dimensional singular perturbation problems based on a petrov galerkin finite element or an equivalent finite volume scheme the method is unique in that its discretization error has a bound that is second order in the mesh size and uniform in the perturbation parameter it satisfies a local discrete conservation law and it exhibits a discrete maximum principle numerical results are included for comparison with the theory

the optimal accuracy of difference schemes we consider difference approximations to the model hyperbolic equation ut ux which compute each new value u x t at as a combination of the known values u x ra x t u x s a x a t for such schemes we find the optimal order of accuracy stability is possible for small a t a x if and only if p min r s 2 r 2 2s a similar bound is established for implicit methods in this case the most accurate schemes are based on pade approximations p z q z to za near z 1 and we find an expression for the difference i q 12 p 12 this allows us to test the von neumann condition i p q i 1 we also determine the number of zeros of q in the unit circle which decides whether the implicit part is uniformly invertible

a riccati transformation method for solving linear bvps i theoretical aspects a method of solving linear two point boundary value problems bvps for first order systems of ordinary differential equations odes is considered of particular interest is the class of singularly perturbed bvps the method attempts to split the integration of the rapidly increasing decreasing solution components typical for such problems into different initial value problems ivps the case of separated boundary conditions bcs is considered and their ability to detect solution components that eventually increase decrease is exploited by using the riccati transformation the original bvp is then reformulated as three ivps one of them being a matrix riccati equation the riccati equation is nonlinear and a multiple embedding setting must be used the entire method is then fit theoretically into a multiple shooting like framework and the original bvp as a chain of subproblems is studied in this way a generalized decoupling result is presented which enables the study of stability to be brough