Mesh Of Shishkin Type Research Articles

Uniformly convergent NIPG method for singularly perturbed convection diffusion problem on Shishkin type meshes

In this paper, we introduce numerical methods for singularly perturbed convection-diffusion problems with a turning point. As a result of the turning point, the problem typically exhibits exponential-type boundary layers or a cusp-type interior layer. We develop non-symmetric discontinuous Galerkin finite element method with interior penalties (NIPG) for both the cases. Usual Shishkin mesh is invoked for the problem with boundary layers whereas generalized Shishkin type mesh is used to tackle the interior layer of cusp-type. The uniform error estimates are obtained for both the problems in L2-norm and DG-norm. Numerical experiments are conducted to confirm the theoretical estimations.

Numerical solution of singularly perturbed Fredholm integro-differential equations by homogeneous second order difference method

This work presents a computational approximate to solve singularly perturbed Fredholm integro-differential equation with the reduced second type Fredholm equation. This problem is discretized by a finite difference approximate, which generates second-order uniformly convergent numerical approximations to the solution. Parameter-uniform approximations are generated using Shishkin type meshes. The performance of the numerical scheme is tested which supports the effectiveness of the technique.

A numerical scheme to solve mixed parabolic-elliptic problem involving singular perturbation

In this article, a singularly perturbed mixed-type problem of parabolic-elliptic is considered on a rectangular domain. The solution of the problem possesses both boundary and interior layers which appears in the spatial variable. To approximate the time derivative, first the implicit Euler scheme and then the Crank–Nicolson scheme on uniform mesh in time direction is used. For approximating the spatial derivative, we use the central difference scheme in the first sub-domain and a hybrid scheme in the second sub-domain on Shishkin type meshes (standard Shishkin mesh and Bakhvalov–Shishkin mesh). We prove that both the numerical schemes converge uniformly with respect to the perturbation parameter and are of second-order accurate. Thomas algorithm is used to solve the tri-diagonal system. The numerical results and the error bounds are illustrated through few tables and graphs.

Convergence analysis of the homogeneous second order difference method for a singularly perturbed Volterra delay-integro-differential equation

• A new approach for designing the computational method has been considered. • This new approach results in a local truncation error containing much lower derivatives of exact solution compared to classical methods. • The method allows to weaken or get rid of the smoothness of the data functions, a determining factor for convergence analysis. • The previous works related to Volterra integro-differential equation were only concerned with regular cases. • Problems involving boundary layers have a solution with bad behaviours in applications encountered in different fields of engineering. A linear Volterra delay-integro-differential equation with a singular perturbation parameter ε is considered. The problem is discretized using exponentially fitted schemes on the Shishkin type meshes. It is proved that the numerical approximations generated by this method are O ( N − 2 ln N ) convergent in the discrete maximum norm, where N is the mesh parameter. Numerical results show a good agreement with the theoretical analysis.

An improved Kellogg-Tsan solution decomposition in numerical methods for singularly perturbed convection-diffusion problems

We consider the Kellogg-Tsan decomposition of the solution to the linear one-dimensional singularly perturbed convection-diffusion problem and improve it by including the solution of the corresponding reduced problem as a component. The upwind scheme on a modified Shishkin-type mesh is used to approximate the unknown component of the decomposition. It is proved that the error is O(ε(ln⁡ε)2N−1), where ε is the perturbation parameter and N is the number of mesh steps. The high accuracy of the method is illustrated by numerical examples.

Superconvergence analysis of FEM and SDFEM on graded meshes for a problem with characteristic layers

We consider a singularly perturbed convection-diffusion boundary value problem whose solution contains exponential and characteristic boundary layers. The problem is numerically solved by the FEM and SDFEM method with bilinear elements on a graded mesh. For the FEM we prove almost uniform convergence and superconvergence. The use of a graded mesh allows for the SDFEM to yield almost uniform estimates in the SD norm, which is not possible for Shishkin type meshes. Numerical results are presented to support theoretical bounds.

Numerical investigation of time delay parabolic differential equation involving two small parameters

PurposeThe purpose of this study is to provide a robust numerical method for a two parameter singularly perturbed delay parabolic initial boundary value problem (IBVP).Design/methodology/approachTo solve the problem, the authors have used a hybrid scheme combining the midpoint scheme, the upwind scheme and the second-order central difference scheme for the spatial derivatives. The backward Euler scheme on a uniform mesh is used to approximate the time derivative. Here, the authors have used Shishkin type meshes for spatial discretization.FindingsIt is observed that the proposed method converges uniformly with almost second-order spatial accuracy with respect to the discrete maximum norm.Originality/valueThis paper deals with the numerical study of a two parameter singularly perturbed delay parabolic IBVP. To solve the problem, the authors have used a hybrid scheme combining the midpoint scheme, the upwind scheme and the second-order central difference scheme for the spatial derivatives. The backward Euler scheme on a uniform mesh is used to approximate the time derivative. The convergence analysis is carried out. It is observed that the proposed method converges uniformly with almost second-order spatial accuracy with respect to the discrete maximum norm. Numerical experiments illustrate the efficiency of the proposed scheme.

Uniformly convergent numerical method for a singularly perturbed differential difference equation with mixed type

In this paper, we deal with the singularly perturbed problem for a linear second order differential difference equation with delay as well as advance. In order to solve the problem numerically, we construct a new difference scheme by the method of integral identities with the use interpolating quadrature rules with remainder terms in integral form. Using an appropriately non-uniform mesh of Shishkin type, we find that the method is almost first order convergent in the discrete maximum norm with respect to the perturbation parameter. Furthermore, we present the numerical experiments that their results support of the theory.

The Bakhvalov mesh: a complete finite-difference analysis of two-dimensional singularly perturbed convection-diffusion problems

A linear two-dimensional singularly perturbed convection-diffusion boundary-value problem is considered. The problem is discretized by the upwind finite-difference method. The analysis of this method on Shishkin-type meshes has been well-established, but the discretization mesh in this paper is the original Bakhvalov mesh, introduced in 1969 as the first layer-adapted mesh. We analyze the error of the numerical solution in the maximum norm and prove first-order pointwise accuracy, uniform in the perturbation parameter. This is the first complete analysis of this kind for two-dimensional convection-diffusion problems discretized on the Bakhvalov mesh. Our numerical experiments validate the theoretical analysis.

A Second Order Weighted Numerical Scheme on Nonuniform Meshes for Convection Diffusion Parabolic Problems

In this article, a singularly perturbed parabolic convection-diffusion equation on a rectangular domain is considered. The solution of the problem possesses regular boundary layer which appears in the spatial variable. To discretize the time derivative, we use two type of schemes, first the implicit Euler scheme and second the implicit trapezoidal scheme on a uniform mesh. For approximating the spatial derivatives, we use the monotone hybrid scheme, which is a combination of midpoint upwind scheme and central difference scheme with variable weights on Shishkin-type meshes (standard Shishkin mesh, Bakhvalov-Shishkin mesh and modified Bakhvalov-Shishkin mesh). We prove that both numerical schemes converge uniformly with respect to the perturbation parameter and are of second order accurate. Thomas algorithm is used to solve the tri-diagonal system. Finally, to support the theoretical results, we present a numerical experiment by using the proposed methods.

Uniformly Convergent Numerical Method for Singularly Perturbed Two Parameter Time Delay Parabolic Problem

This paper discusses the numerical solution of one dimensional parabolic convection–reaction–diffusion time delay problem with two small parameters. For the discretization of the time derivative, we use the implicit Euler scheme on a uniform mesh and for the spatial discretization, we use the upwind difference scheme on the Shishkin type meshes (standard Shishkin mesh, Bakhvalov–Shishkin mesh). We prove that numerically the proposed method is provides a first order convergence, which is optimal for this case. Finally, to support the theoretical results, we present some numerical experiments using the proposed method.

A uniformly convergent hybrid difference scheme for a system of singularly perturbed initial value problems

ABSTRACTA system of second-order singularly perturbed initial value problems with weak hypotheses on the coefficients is considered. The equations have diffusion parameters of different magnitudes, which gives rise to overlapping boundary layers. The structure of these layers is analysed, and this leads to the construction of a Shishkin-type mesh. On this mesh a hybrid difference scheme is proposed, which is a combination of the second-order difference schemes on the fine mesh and the midpoint upwind scheme on the coarse mesh. By applying the truncation error estimate techniques and a difference analogue of Gronwall's inequality, it is proved that the scheme is almost second-order convergent, in the discrete maximum norm, independently of singular perturbation parameters. Numerical experiments are provided to validate the theoretical results.

Uniform Convergence Method for a Delay Differential Problem with Layer Behaviour

Difference method on a piecewise uniform mesh of Shishkin type, for a singularly perturbed boundary-value problem for a linear second-order delay differential equation is examined. It is proved that it gives essentially a first-order parameter-uniform convergence in the discrete maximum norm. Furthermore, numerical results are presented in support of the theory.

Uniformly Convergent Numerical Method for Singularly Perturbed Time Delay Parabolic Problem with Two Small Parameters

This article discusses the numerical solution of one dimensional parabolic convection-reaction-diffusion time delay problem with two small parameters. For the discretization of the time derivative, we use the implicit Euler scheme on a uniform mesh and for the spatial discretization, we use the upwind difference scheme on the Shishkin type meshes (standard Shishkin mesh, Bakhvalov–Shishkin mesh). We prove that the proposed method is uniformly convergent, parameter independent and provides a first order convergence, which is optimal for this case. Finally, to support the theoretical results, we present some numerical experiments using the proposed method.

A second order numerical method for singularly perturbed delay parabolic partial differential equation

Purpose The purpose of this paper is to provide a robust second-order numerical scheme for singularly perturbed delay parabolic convection–diffusion initial boundary value problem. Design/methodology/approach For the parabolic convection-diffusion initial boundary value problem, the authors solve the problem numerically by discretizing the domain in the spatial direction using the Shishkin-type meshes (standard Shishkin mesh, Bakhvalov–Shishkin mesh) and in temporal direction using the uniform mesh. The time derivative is discretized by the implicit-trapezoidal scheme, and the spatial derivatives are discretized by the hybrid scheme, which is a combination of the midpoint upwind scheme and central difference scheme. Findings The authors find a parameter-uniform convergent scheme which is of second-order accurate globally with respect to space and time for the singularly perturbed delay parabolic convection–diffusion initial boundary value problem. Also, the Thomas algorithm is used which takes much less computational time. Originality/value A singularly perturbed delay parabolic convection–diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a hybrid scheme. The method is parameter-uniform convergent and is of second order accurate globally with respect to space and time. Numerical results are carried out to verify the theoretical estimates.

Higher Order Uniformly Convergent Numerical Algorithm for Time-Dependent Singularly Perturbed Differential-Difference Equations

This paper presents a higher order uniformly convergent discretization for second order singularly perturbed parabolic differential equation with delay and advance terms. The retarded terms are approximated by the Taylor series which give rise to a nearby singularly perturbed parabolic differential equation. A hybrid numerical algorithm based on implicit Euler scheme is proposed to discretize the time variable and a combined finite difference scheme made out of modified upwind and central difference schemes to discretize the spatial variable on a piecewise uniform mesh of Shishkin type in space and a uniform mesh in time. The existence and uniqueness of a solution for the proposed hybrid algorithm is analyzed. It is proved that the algorithm is $$\epsilon $$ -uniformly convergent of almost second order in space, $$O(N^{-2} (\ln N)^2)$$ and first order in time, $$O(M^{-1})$$ . A few numerical experiments supporting the results are presented. The efficiency of the proposed hybrid algorithm is demonstrated by comparing upwind and modified upwind algorithm.

Analysis of the truncation error and barrier-function technique for a Bakhvalov-type mesh

We use a barrier-function technique to prove the parameter-uniform convergence for singularly perturbed convection-diffusion problems discretized on a Bakhvalov-type mesh. This is the first proof of this kind in the research literature, the barrier-function approach having only been applied so far to Shishkin-type meshes.

A robust fitted numerical method for singularly perturbed turning point problems whose solution exhibits an interior layer

In this paper we considerer singularly perturbed convection-diffusion-reaction problems with a turning point whose solution exhibits an interior layer. We establish bounds on the solution to these problems and their derivatives. We construct a fitted mesh finite difference method (FMFDM). The method consists of an upwind scheme on an appropriately designed piecewise uniform mesh of Shishkin type. This mesh is fine near the turning point and coarse elsewhere. A rigorous error analysis shows that the developed method is uniformly convergent of order almost one. In order to improve the accuracy of the proposed FMFDM, we apply Richardson extrapolation. Two numerical examples are considered to illustrate the theoretical findings. Results show that this method is reliable and competitive.

An efficient numerical method for singularly perturbed time dependent parabolic 2D convection–diffusion systems

In this paper we deal with solving efficiently 2D linear parabolic singularly perturbed systems of convection–diffusion type. We analyze only the case of a system of two equations where both of them feature the same diffusion parameter. Nevertheless, the method is easily extended to systems with an arbitrary number of equations which have the same diffusion coefficient. The fully discrete numerical method combines the upwind finite difference scheme, to discretize in space, and the fractional implicit Euler method, together with a splitting by directions and components of the reaction–convection–diffusion operator, to discretize in time. Then, if the spatial discretization is defined on an appropriate piecewise uniform Shishkin type mesh, the method is uniformly convergent and it is first order in time and almost first order in space. The use of a fractional step method in combination with the splitting technique to discretize in time, means that only tridiagonal linear systems must be solved at each time level of the discretization. Moreover, we study the order reduction phenomenon associated with the time dependent boundary conditions and we provide a simple way of avoiding it. Some numerical results, which corroborate the theoretical established properties of the method, are shown.

NUMERICAL SOLUTION OF A BOUNDARY VALUE PROBLEM INCLUDING BOTH DELAY AND BOUNDARY LAYER

Difference method on a piecewise uniform mesh of Shishkin type, for a singularly perturbed boundary-value problem for a nonlinear second order delay differential equation is analyzed. Also, the method is proved that it gives essentially first order parameter-uniform convergence in the discrete maximum norm. Furthermore, numerical results are presented in support of the theory.