Fourth-order finite-difference method for boundary value problems with two small parameters

We present a finite difference scheme for a class of linear singularly perturbed boundary value problems with two small parameters. The problem is discretized using a Bakhvalov-type mesh. It is proved under certain conditions that this scheme is fourth-order accurate and that its error does not increase when the perturbation parameter tends to zero. Numerical examples are presented which demonstrate computationally the fourth order of the method.     

Numerical solution of a mixed singularly perturbed parabolic-elliptic problem

A one-dimensional singularly perturbed problem of mixed type is considered. The domain under consideration is partitioned into two subdomains. In the first subdomain a parabolic reaction-diffusion problem is given and in the second one an elliptic convection-diffusion-reaction problem. The solution is decomposed into regular and singular components. The problem is discretized using an inverse-monotone finite volume method on condensed Shishkin meshes. We establish an almost second-order global pointwise convergence in the space variable, that is uniform with respect to the perturbation parameter.     

Analysis of a FEM for a coupled system of singularly perturbed reaction–diffusion equations

We consider a system of ℓ ≥ 2 one-dimensional singularly perturbed reaction–diffusion equations coupled at the zero-order term. The second derivative of each equation is multiplied by a distinct small parameter. We present a convergence theory for conforming linear finite elements on arbitrary meshes. As a result convergence independently of the perturbation parameters on a wide class of layer-adapted meshes is established.      

Numerical solution of singularly perturbed convection-diffusion problem using parameter uniform B-spline collocation method

This paper is concerned with a numerical scheme to solve a singularly perturbed convection-diffusion problem. The solution of this problem exhibits the boundary layer on the right-hand side of the domain due to the presence of singular perturbation parameter ε. The scheme involves B-spline collocation method and appropriate piecewise-uniform Shishkin mesh. Bounds are established for the derivative of the analytical solution. Moreover, the present method is boundary layer resolving as well as second-order uniformly convergent in the maximum norm. A comprehensive analysis has been given to prove the uniform convergence with respect to singular perturbation parameter. Several numerical examples are also given to demonstrate the efficiency of B-spline collocation method and to validate the theoretical aspects.     

A defect–correction parameter-uniform numerical method for a singularly perturbed convection–diffusion problem in one dimension

Monotone second order parameter-robust numerical methods for singularly perturbed differential equations can be designed using the principles of defect-correction. However, the proofs of second order parameter-uniform convergence can be difficult, even in one dimension. In this paper, we examine a variant of the standard defect-correction scheme. This variant is proposed in order to simplify the analysis of a defect-correction method in the case of a one dimensional convection–diffusion problem. Numerical results are presented to validate the theoretical results.*This research was partially supported by the project MEC/FEDER MTM 2004-019051 and the grant EUROPA XXI of the Caja de Ahorros de la Inmaculada.     

Scaled discrete derivatives of singularly perturbed elliptic problems

Numerical approximations to the solution of a singularly perturbed elliptic convection–diffusion problem in two space dimensions are generated using a monotone finite difference operator on a tensor product of piecewise‐uniform Shishkin meshes. The bilinear interpolants of these numerical approximations are parameter‐uniformly convergent to the solution of the continuous problem, in the pointwise maximum norm. In this article, discrete approximations to the first derivatives of the solution are shown to be globally first‐order (up to logarithmic factors) uniformly convergent, when the errors are scaled within the analytical layers of the continuous problem. Numerical results are presented to illustrate the theoretical error bounds established in an appropriated weighted C¹–norm. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 225–252, 2015     

Higher order global solution and normalized flux for singularly perturbed reaction-diffusion problems

In this paper a computational technique is proposed for obtaining a higher order global solution and global normalized flux of singularly perturbed reaction-diffusion two-point boundary-value problems. The HOC (higher order compact) finite difference scheme developed in Gracia et al. (2001) [4] and which is constructed on an appropriate piecewise uniform Shishkin mesh, has been considered to find an almost fourth order convergent solution at mesh points. Using these values, piecewise cubic interpolants based approximations for solution and normalized flux in whole domain have been defined. It has been shown that the global solution and the global normalized flux are also uniformly convergent. Moreover, for the global solution, the order of uniform convergence in the whole domain is optimal, i.e., it is the same as this one obtained at mesh points, whereas, for the global normalized flux, the uniform convergence is almost third order, except at midpoints of the mesh, where it is also almost fourth order. Theoretical error bounds have been provided along with some numerical examples, which corroborate the efficiency of the proposed technique to find good approximations to the global solution and the global normalized flux.     

Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem

A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter is arbitrarily small, which induces boundary layers. Constructing discrete sub- and super-solutions, we prove existence and investigate the accuracy of multiple discrete solutions on layer-adapted meshes of Bakhvalov and Shishkin types. It is shown that one gets second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the discrete maximum norm, uniformly in for . Here is the maximum side length of mesh elements, while the number of mesh nodes does not exceed . Numerical experiments are performed to support the theoretical results.

     

Uniformly convergent difference schemes for a singularly perturbed third order boundary value problem

In this paper we consider a numerical approximation of a third order singularly perturbed boundary value problem by an upwind finite difference scheme on a Shishkin mesh. The behavior of the solution, and the stability of the continuous problem are discussed. The proof of the uniform convergence of the proposed numerical method is based on the strongly uniform stability and a weak consistency property of the discrete problem. Numerical experiments verify our theoretical results.     

A uniformly convergent B-spline collocation method on a nonuniform mesh for singularly perturbed one-dimensional time-dependent linear convection–diffusion problem

A numerical method is proposed for solving singularly perturbed one-dimensional parabolic convection–diffusion problems. The method comprises a standard implicit finite difference scheme to discretize in temporal direction on a uniform mesh by means of Rothe's method and B-spline collocation method in spatial direction on a piecewise uniform mesh of Shishkin type. The method is shown to be unconditionally stable and accurate of order O((Δx)²+Δt). An extensive amount of analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter. Several numerical experiments have been carried out in support of the theoretical results. Comparisons of the numerical solutions are performed with an upwind finite difference scheme on a piecewise uniform mesh and exponentially fitted method on a uniform mesh to demonstrate the efficiency of the method.     

Layers and corner singularities in singularly perturbed elliptic problems

Some recent work and open problems are reviewed concerning the numerical solution of singularly perturbed elliptic boundary value problems whose solutions have boundary layers and corner singularities. AMS subject classification (2000)  65N15, 65N50, 35J25     

The singularly perturbed problem for combustion reaction diffusion

1. IntroductionThe anchor discussed a class of problems for the nonlinear ordinary differential equations(of. [1-3]) and the reaction dimsion equations (of. [4-7]).Now we consider a model of the combustion reaction decision problem as follows:where y is a density of combustion flame, s denotes the rate of diffusion effect for reactionspeed, whiCh is a small positive parameter, t is a position of combustion, functions y - t andy t are the positions of the matter for fuel and chide respectivel…     

Numerical method for a singularly perturbed convection-diffusion problem with delay

This paper deals with the singularly perturbed boundary value problem for a linear second-order delay differential equation. For the numerical solution of this problem, we use an exponentially fitted difference scheme on a uniform mesh which is accomplished by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form. It is shown that one gets first order convergence in the discrete maximum norm, independently of the perturbation parameter. Numerical results are presented which illustrate the theoretical results.     

High order parameter-robust numerical method for singularly perturbed reaction-diffusion problems

We present a high order parameter-robust finite difference method for singularly perturbed reaction-diffusion problems. The problem is discretized using a suitable combination of fourth order compact difference scheme and central difference scheme on generalized Shishkin mesh. The convergence analysis is given and the method is proved to be almost fourth order uniformly convergent in maximum norm with respect to singular perturbation parameter ε. Numerical experiments are conducted to demonstrate the theoretical results.     

High order robust approximations for singularly perturbed semilinear systems

In this paper, a system of M (?2) singularly perturbed semilinear reaction–diffusion equations is considered. To obtain a high order approximation to the solution of this system, we propose a hybrid numerical method that employs a generalized Shishkin mesh with the Numerov discretization in the boundary layer regions and either a non-equidistant generalization of the Numerov discretization or classical central differences in the outer region. It is proved that the method is almost fourth order convergent in the maximum norm uniformly with respect to the perturbation parameter. Numerical experiments support the theoretical results and demonstrate the effectiveness of the method.     

Superconvergence analysis of the Galerkin FEM for a singularly perturbed convection–diffusion problem with characteristic layers

We analyze the superconvergence property of the Galerkin finite element method (FEM) for elliptic convection–diffusion problems with characteristic layers. This method on Shishkin meshes is known to be almost first‐order accurate (up to a logarithmic factor) in the energy norm induced by the bilinear form of the weak formulation, uniformly in the perturbation parameter. In the present paper the method is shown to be almost second‐order superconvergent in this energy norm for the difference between the FEM solution and the bilinear interpolant of the exact solution. This supercloseness property is used to improve the accuracy to almost second order by means of a postprocessing procedure. Numerical experiments confirm these results. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

An alternating direction scheme on a nonuniform mesh for reaction-diffusion parabolic problems

In this paper we develop a numerical method for two-dimensionaltime-dependent reaction-diffusion problems. This method, whichcan immediately be generalized to higher dimensions, is shownto be uniformly convergent with respect to the diffusion problems.This method, which can immediately be generalized to higherdimensions, is shown to be uniformly convergent with respectto the diffusion parameter.     

An almost third order finite difference scheme for singularly perturbed reaction-diffusion systems

This paper addresses the numerical approximation of solutions to coupled systems of singularly perturbed reaction-diffusion problems. In particular a hybrid finite difference scheme of HODIE type is constructed on a piecewise uniform Shishkin mesh. It is proved that the numerical scheme satisfies a discrete maximum principle and also that it is third order (except for a logarithmic factor) uniformly convergent, even for the case in which the diffusion parameter associated with each equation of the system has a different order of magnitude. Numerical examples supporting the theory are given.     

Solving a partially singularly perturbed initial value problem on Shishkin meshes

A coupled first order system of one singularly perturbed and one non-perturbed ordinary differential equation with prescribed initial conditions is considered. A Shishkin piecewise uniform mesh is constructed and used, in conjunction with a classical finite difference operator, to form a new numerical method for solving this problem. It is proved that the numerical approximations generated by this method are essentially first order convergent in the maximum norm at all points of the domain, uniformly with respect to the singular perturbation parameter. Numerical results are presented in support of the theory.     

A parameter‐uniform scheme for the parabolic singularly perturbed problem with a delay in time

In this paper, a parameter‐uniform numerical scheme for the solution of singularly perturbed parabolic convection–diffusion problems with a delay in time defined on a rectangular domain is suggested. The presence of the small diffusion parameter ? leads to a parabolic right boundary layer. A collocation method consisting of cubic B ‐spline basis functions on an appropriate piecewise‐uniform mesh is used to discretize the system of ordinary differential equations obtained by using Rothe's method on an equidistant mesh in the temporal direction. The parameter‐uniform convergence of the method is shown by establishing the theoretical error bounds. The numerical results of the test problems validate the theoretical error bounds.