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.     

On adaptive mesh for the initial boundary value singularly perturbed delay Sobolev problems

A uniform finite difference method on a B-mesh is applied to solve the initial-boundary value problem for singularly perturbed delay Sobolev equations. To solve the foresold problem, finite difference scheme on a special nonuniform mesh, whose solution converges point-wise independently of the singular perturbation parameter is constructed and analyzed. The present paper also aims at discussing the stability and convergence analysis of the method. An error analysis shows that the method is of second order convergent in the discrete maximum norm independent of the perturbation parameter. A numerical example and the simulation results show the effectiveness of our theoretical results.     

A singular perturbation approach to solve Burgers–Huxley equation via monotone finite difference scheme on layer-adaptive mesh

This paper deals with a numerical method for solving one-dimensional unsteady Burgers–Huxley equation with the viscosity coefficient ε. The parameter ε takes any values from the half open interval (0, 1]. At small values of the parameter ε, an outflow boundary layer is produced in the neighborhood of right part of the lateral surface of the domain and the problem can be considered as a non-linear singularly perturbed problem with a singular perturbation parameter ε. Using singular perturbation analysis, asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular components. We construct a numerical scheme that comprises of implicit-Euler method to discretize in temporal direction on uniform mesh and a monotone hybrid finite difference operator to discretize the spatial variable with piecewise uniform Shishkin mesh. To obtain better accuracy, we use central finite difference scheme in the boundary layer region. Shishkin meshes are refined in the boundary layer region, therefore stability constraint is satisfied by proposed scheme. Quasilinearization process is used to tackle the non-linearity and it is shown that quasilinearization process converges quadratically. The method has been shown to be first order uniformly accurate in the temporal variable, and in the spatial direction it is first order parameter uniform convergent in the outside region of boundary layer, and almost second order parameter uniform convergent in the boundary layer region. Accuracy and uniform convergence of the proposed method is demonstrated by numerical examples and comparison of numerical results made with the other existing methods.     

A numerical method for singularly perturbed weakly coupled system of two second order ordinary differential equations with discontinuous source term

In this paper, a numerical method based on finite difference scheme and Shishkin mesh for singularly perturbed two second order weakly coupled system of ordinary differential equations with discontinuous source term is presented. An error estimate is derived to show that the method is uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented to illustrate the theoretical results.     

A second order scheme for a time-dependent,singularly perturbed convection–diffusion equation

We consider a numerical scheme for a one-dimensional, time-dependent, singularly perturbed convection–diffusion problem. The problem is discretized in space by a standard finite element method on a Bakhvalov–Shishkin type mesh. The space error is measured in an L² norm. For the time integration, the implicit midpoint rule is used. The fully discrete scheme is shown to be convergent of order 2 in space and time, uniformly in the singular perturbation parameter.     

Convergence Analysis of Weighted Difference Approximations on Piecewise Uniform Grids to a Class of Singularly Perturbed Functional Differential Equations

We consider the numerical approximation of a singularly perturbed time delayed convection diffusion problem on a rectangular domain. Assuming that the coefficients of the differential equation be smooth, we construct and analyze a higher order accurate finite difference method that converges uniformly with respect to the singular perturbation parameter. The method presented is a combination of the central difference spatial discretization on a Shishkin mesh and a weighted difference time discretization on a uniform mesh. A?priori explicit bounds on the solution of the problem are established. These bounds on the solution and its derivatives are obtained using a suitable decomposition of the solution into regular and layer components. It is shown that the proposed method is $L_{2}^{h}$ -stable. The analysis done permits its extension to the case of adaptive meshes which may be used to improve the solution. Numerical examples are presented to demonstrate the effectiveness of the method. The convergence obtained in practical satisfies the theoretical predictions.     

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.     

On piecewise-uniform meshes for upwind- and central-difference operators for solving singularly perturbed problems

Numerical methods composed of upwind-difference operators onuniform meshes are shown analytically to be defective for solvingsingularly perturbed differential equations, in the sense that,as the mesh is refined, the error in the numerical approximationincreases until the mesh parameter has decreased to the sameorder of magnitude as the singular perturbation parameter. Itis also shown that the same is true for upwind-difference operatorson piecewise uniform meshes having a transition point that isdependent solely on the singular perturbation parameter. Itis then shown, with specific analytic examples, that both upwind-and central-difference operators on specially designed piecewise-uniformmeshes give numerical methods which do not suffer from thisdefect. Conditions are also given on the structure of a piecewiseuniform mesh that are necessary if the numerical method, composedof this mesh and an upwind-difference operator, is to be convergentuniformly with respect to the singular perturbation parameter.     

A parameter-uniform numerical method for time-dependent singularly perturbed differential-difference equations

A numerical study is made for solving a class of time-dependent singularly perturbed convection–diffusion problems with retarded terms which often arise in computational neuroscience. To approximate the retarded terms, a Taylor’s series expansion has been used and the resulting time-dependent singularly perturbed differential equation is approximated using parameter-uniform numerical methods comprised of a standard implicit finite difference scheme to discretize in the temporal direction on a uniform mesh by means of Rothe’s method and a B-spline collocation method in the spatial direction on a piecewise-uniform mesh of Shishkin type. The method is shown to be accurate of order O(M⁻¹ + N⁻² ln³N), where M and N are the number of mesh points used in the temporal direction and in the spatial direction respectively. An extensive amount of analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter. Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations. Comparisons of the numerical solutions are performed with an upwind and midpoint upwind finite difference scheme on a piecewise-uniform mesh to demonstrate the efficiency of the method.     

A layer adaptive B‐spline collocation method for singularly perturbed one‐dimensional parabolic problem with a boundary turning point

In this article, we develop a parameter uniform numerical method for a class of singularly perturbed parabolic equations with a multiple boundary turning point on a rectangular domain. The coefficient of the first derivative with respect to x is given by the formula a₀(x, t)x^p, where a₀(x, t) ≥ α > 0 and the parameter p ∈ [1,∞) takes the arbitrary value. For small values of the parameter ε, the solution of this particular class of problem exhibits the parabolic boundary layer in a neighborhood of the boundary x = 0 of the domain. We use the implicit Euler method to discretize the temporal variable on uniform mesh and a B‐spline collocation method defined on piecewise uniform Shishkin mesh to discretize the spatial variable. Asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular component. These bounds are applied in the convergence analysis of the proposed scheme on Shishkin mesh. The resulting method is boundary layer resolving and has been shown almost second‐order accurate in space and first‐order accurate in time. It is also shown that the proposed method is uniformly convergent with respect to the singular perturbation parameter ε. Some numerical results are given to confirm the predicted theory and comparison of numerical results made with a scheme consisting of a standard upwind finite difference operator on a piecewise uniform Shishkin mesh. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1143–1164, 2011     

A compact finite difference scheme for 2D reaction–diffusion singularly perturbed problems

In this work we define a compact finite difference scheme of positive type to solve a class of 2D reaction–diffusion elliptic singularly perturbed problems. We prove that if the new scheme is constructed on a piecewise uniform mesh of Shishkin type, it provides better approximations than the classical central finite difference scheme. Moreover, the uniform parameter bound of the error shows that the scheme is third order convergent in the maximum norm when the singular perturbation parameter is sufficiently small. Some numerical experiments illustrate in practice the result of convergence proved theoretically.     

Fitted finite difference method for singularly perturbed delay differential equations

This paper deals with singularly perturbed initial value problem for linear second-order delay differential equation. An exponentially fitted difference scheme is constructed in an equidistant mesh, which gives first order uniform convergence in the discrete maximum norm. The difference scheme is shown to be uniformly convergent to the continuous solution with respect to the perturbation parameter. A numerical example is solved using the presented method and compared the computed result with exact solution of the problem.     

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.     

Uniform convergence analysis of hybrid numerical scheme for singularly perturbed problems of mixed type

In this article, we consider a class of singularly perturbed mixed parabolic‐elliptic problems whose solutions possess both boundary and interior layers. To solve these problems, a hybrid numerical scheme is proposed and it is constituted on a special rectangular mesh which consists of a layer resolving piecewise‐uniform Shishkin mesh in the spatial direction and a uniform mesh in the temporal direction. The domain under consideration is partitioned into two subdomains. For the spatial discretization, the proposed scheme is comprised of the classical central difference scheme in the first subdomain and a hybrid finite difference scheme in the second subdomain, whereas the time derivative in the given problem is discretized by the backward‐Euler method. We prove that the method converges uniformly with respect to the perturbation parameter with almost second‐order spatial accuracy in the discrete supremum norm. Numerical results are finally presented to validate the theoretical results.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1931–1960, 2014     

A finite difference scheme for a class of singularly perturbed initial value problems for delay differential equations

This study deals with the singularly perturbed initial value problem for a quasilinear first-order delay differential equation. A numerical method is generated on a grid that is constructed adaptively from a knowledge of the exact solution, which involves appropriate piecewise-uniform mesh on each time subinterval. An error analysis shows that the method is first order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. The parameter uniform convergence is confirmed by numerical computations.     

An ?-uniform hybrid scheme for singularly perturbed delay differential equations

In this paper, a singularly perturbed delay differential equation of first order has been considered. The problem is solved by using a hybrid scheme on a Shishkin mesh. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter. Truncation errors are obtained. Finally, numerical experiments are carried out on a test problem, confirming the effectiveness of the proposed technique.     

Optimal error estimate of upwind scheme on Shishkin-type meshes for singularly perturbed parabolic problems with discontinuous convection coefficients

This paper analyzes the implicit upwind finite difference scheme on Shishkin-type meshes (including the classical piecewise-uniform Shishkin mesh and the Bakhalov-Shishkin mesh) for a class of singularly perturbed parabolic convection-diffusion problems exhibiting strong interior layers. Suitable conditions on the mesh-generating functions are derived and are found to be sufficient for the convergence of the method, uniformly with respect to the perturbation parameter. Utilizing these conditions, it is shown that the method converges uniformly in the discrete supremum norm with an optimal error bound. Numerical results are presented to validate the theoretical results.     

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.     

A posteriori error analysis for defect correction method for two parameter singular perturbation problems

Defect correction method is used for two parameter singular perturbation problem on Bakhvalov-Shishkin mesh. Use of defect correction method on Bakhvalov-Shishkin mesh gives a second order convergence. A posteriori error estimate is obtained. The numerical examples are given to establish the second order convergence in practice.     

On the hybrid finite difference scheme for a singularly perturbed Riccati equation

In this paper a singularly perturbed Riccati equation is considered. The problem is solved by using a hybrid finite difference method on a Shishkin mesh. The method is composed of the midpoint upwind scheme and the backward Euler scheme based on the relation between the local mesh width and the perturbation parameter. It is proved that the scheme is almost second-order convergent, in the global maximum norm, independently of the singular perturbation parameter. Numerical experiments support the theoretical results.