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We develop a local discontinuous Galerkin (LDG) method for a fourth-order singularly perturbed problem of convection-diffusion type. The existence and uniqueness of the computed solution are verified. Using the Shishkin mesh we derive an optimal-order energy-norm error estimate which is uniformly valid in the perturbation parameter. Numerical experiments are also given to support our theoretical findings. [ABSTRACT FROM AUTHOR]

This paper provides a qualitative and quantitative study of a second-order Singularly Perturbed Reaction–Diffusion type System of Integro-differential equations with discontinuous source term. To obtain the numerical solution of the problem, an exponentially-fitted method that can be applied to a Shishkin mesh. This method shows that uniform convergence with respect to the perturbation parameter and necessary examples are given. [ABSTRACT FROM AUTHOR]

This article presents a numerical solution for a specific class of 2D parabolic singularly perturbed convection-diffusion problems with a special interior line source. The proposed approach employs the alternating direction implicit type operator splitting streamline-diffusion finite element method (SDFEM), offering a viable solution to alleviate computational complexity and high storage requirements encountered in higher-dimensional problems. The overall stability of the two-step method is established, while a piecewise-uniform Shishkin mesh is employed for spatial domain discretization. By carefully selecting the stabilization parameter, an ε -uniform error estimate is derived, accounting for the influence of the time step-interval which is essential to maintain the method’s stability. To validate the theoretical error estimate, numerical investigation are conducted, showcasing the effectiveness of the proposed method. This research contributes to advancing the understanding and numerical treatment of this specific class of 2D parabolic singularly perturbed convection-diffusion problems, shedding light on the intricate dynamics and behavior of the system in the presence of a special interior line source. [ABSTRACT FROM AUTHOR]

In this paper, we propose a weak Galerkin finite element approximation for a class of fourth-order singularly perturbed parabolic problems. The problem exhibits boundary layers and so we have considered layer adapted triangulations, in particular Shishkin triangular mesh in the spatial domain. For temporal discretization, we utilize the Crank-Nicolson scheme on a uniform mesh. Stability and error estimates along with the uniform convergence of the method has been proved. Numerical examples are included which verifies our analysis. [ABSTRACT FROM AUTHOR]

The solution of the singular perturbation problems (SPP) of convection-diffusion-reaction type may exhibit regular and corner layers in a rectangular domain. In this work, we construct and analyze a parameter-uniform operator-splitting alternating direction implicit (ADI) scheme to efficiently solve a two-dimensional parabolic singularly perturbed problem with two positive parameters. The proposed model is a combination of the backward-Euler method defined on a uniform mesh in time and a hybrid method in space defined on a special Shishkin mesh. The analysis is presented on a layer adapted piecewise-uniform Shishkin mesh. The developed numerical method is proved to be first-order convergent in time and almost second-order convergent in space. The numerical experiments are performed to validate the theoretical convergence results and illustrate the efficiency of the current strategy. [ABSTRACT FROM AUTHOR]

This article focuses on developing and analyzing an efficient higher-order numerical approximation of singularly perturbed two-dimensional semilinear parabolic convection-diffusion problems with time-dependent boundary conditions. We approximate the governing nonlinear problem by an implicit fitted mesh method (FMM), which combines an alternating direction implicit scheme in the temporal direction together with a higher-order finite difference scheme in the spatial directions. Since the solution possesses exponential boundary layers, a Cartesian product of piecewise-uniform Shishkin meshes is used to discretize in space. To begin our analysis, we establish the stability corresponding to the continuous nonlinear problem, and obtain a-priori bounds for the solution derivatives. Thereafter, we pursue the stability analysis of the discrete problem, and prove ε -uniform convergence in the maximum-norm. Next, for enhancement of the temporal accuracy, we use the Richardson extrapolation technique solely in the temporal direction. In addition, we investigate the order reduction phenomenon naturally occurring due to the time-dependent boundary data and propose a suitable approximation to tackle this effect. Finally, we present the computational results to validate the theoretical estimates. [ABSTRACT FROM AUTHOR]

In this article, we explore the discontinuous Galerkin finite element method for two-parametric singularly perturbed convection-diffusion problems with a discontinuous source term. Due to the discontinuity in the source term, the problem typically shows a weak interior layer. Also, the presence of multiple perturbation parameters in the problem causes boundary layers on both sides of the boundary. In this work, we develop the nonsymmetric discontinuous Galerkin finite element method with interior penalties to handle the layer phenomenon. With the use of a typical Shishkin mesh, the domain is discretized, and a uniform error estimate is obtained. Numerical experiments are conducted to validate the theoretical conclusions.

A singularly perturbed reaction–diffusion problem in 1D is solved numerically by a local discontinuous Galerkin (LDG) finite element method. For this type of problem the standard energy norm is too weak to capture the contribution of the boundary layer component of the true solution, so balanced norms have been used by many authors to give more satisfactory error bounds for solutions computed using various types of finite element method. But for the LDG method, up to now no optimal-order balanced-norm error estimate has been derived. In this paper, we consider an LDG method with central numerical flux on a Shishkin mesh. Using the superconvergence property of the local L 2 projector and some local coupled projections around the two transition points of the mesh, we prove an optimal-order balanced-norm error estimate for the computed solution; that is, when piecewise polynomials of degree k are used on a Shishkin mesh with N mesh intervals, in the balanced norm we establish O ((N - 1 ln N) k + 1) convergence when k is even and O ((N - 1 ln N) k) when k is odd. Numerical experiments confirm the sharpness of these error bounds. [ABSTRACT FROM AUTHOR]

This paper is contributed to explore how a Crank-Nicolson weak Galerkin finite element method (WG-FEM) addresses the singularly perturbed unsteady convection-diffusion equation with a nonlinear reaction term in 2D. The problem and some asymptotic behavior results are given for the exact solution and its derivatives with the parameter ε. These results are essential for proving the uniform convergence of the proposed WG-FEM. A tensor product Shishkin mesh featuring piece-wise discontinuous bilinear polynomials is employed to handle the layers for uniform convergence at different parameter values of ε. An error estimate ‖ u N − I N u ‖ W G is presented, where I N u denotes the vertex-edge-cell interpolation of the solution u , and ‖ ⋅ ‖ W G denotes the Weak Galerkin norm. The optimal uniform convergence order is demonstrated for semi-discrete and fully discrete schemes. Various numerical experiments are conducted to validate the optimal order of convergence demonstrated by the proposed method. [ABSTRACT FROM AUTHOR]

A Petrov-Galerkin finite element method is constructed for a singularly perturbed elliptic problem in two space dimensions. The solution contains a regular boundary layer and two characteristic boundary layers. Exponential splines are used as test functions in one coordinate direction and are combined with bilinear trial functions defined on a Shishkin mesh. The resulting numerical method is shown to be a stable parameter-uniform numerical method that achieves a higher order of convergence compared to upwinding on the same mesh. [ABSTRACT FROM AUTHOR]

In this article, a multiscale nonlinear system of singularly perturbed differential equations of convection-diffusion type is considered. A numerical technique combined with the continuation method is constructed to obtain the numerical computations. The newly developed numerical method is shown to be first order convergent uniformly with respect to the perturbation parameter. [ABSTRACT FROM AUTHOR]

In this article, a nonlinear system of singularly perturbed differential equations of convectiondiffusion type with Dirichlet boundary conditions is considered on the interval [0,1]. Both components of the solution of the system exhibit boundary layers near t = 0. A new computational method involving classical finite difference operators, a piecewise-uniform Shishkin mesh and an algorithm based on the continuation method is developed to compute the numerical approximations. The computational method is proved to be first order convergent uniformly with respect to the perturbation parameters. Numerical experiments support the theoretical results. [ABSTRACT FROM AUTHOR]

This study deals with singularly perturbed Volterra integro-differential equations with delay. Based on the properties of the exact solution, a hybrid difference scheme with appropriate quadrature rules on a Shishkin-type mesh is constructed. By using the truncation error estimate techniques and a discrete analogue of Grönwall's inequality it is proved that the hybrid finite difference scheme is almost second order accurate in the discrete maximum norm. Numerical experiments support these theoretical results and indicate that the estimates are sharp. [ABSTRACT FROM AUTHOR]

A singularly perturbed transport equation is considered. A variable two-step backward differentiation formulas (BDF2) on a Shishkin-type mesh is used to discrete the first-order derivatives of the singularly perturbed transport equation. The stability and error analysis are derived by using the discrete orthogonal convolution kernels. It is proved that the scheme is second-order uniformly convergent with respect to the small parameter, which improves previous results. Numerical experiments are presented to support the theoretical result. [ABSTRACT FROM AUTHOR]

Nonlinear science plays a vital role in modern technology. Due to the limitations over the linear theories and the chaotic nature of the problems in this technological era, the analysis of nonlinear problems has become extremely essential to investigate the dynamics of complicated and multiscale characteristics systems. In this article, a nonlinear system of singularly perturbed differential equations with Dirichlet boundary conditions is considered. Both components of the system exhibit a strong boundary layer near the boundary t = 0 of the domain, even though a perturbation parameter occurs only in one of the equations of the system. A robust and layer-resolving numerical method involving the continuation technique is constructed to obtain the numerical approximations to the system. The newly developed numerical method is shown to be first-order convergent uniformly with respect to the perturbation parameter. [ABSTRACT FROM AUTHOR]

The present study investigates an algorithm numerically for finding the solution of partial differential equation with differences involved singular perturbation parameter(SPPDE) on non-uniform grid. Taylor series expansion provides a close approximation of the delay and advance terms in the convection-diffusion terms. After the approximations in shift containing terms, we applied the Crank-Nicolson application on uniform grid in the vertical direction. Subsequently, the resultant system is employed by the method of tension spline on a piece-wise uniform grid. Empirical evidence has shown that the suggested approach exhibits second-order characteristics in both the spatial and temporal dimensions. The effectiveness of derived scheme demonstrated through the solution of examples and the results are compared with existed methods. In the conclusion section, we will discuss the effect of shift parameters behavior for various singular perturbation parameter. [ABSTRACT FROM AUTHOR]

In this work, a compact finite difference approach is constructed for singularly perturbed parabolic reaction diffusion problems with a retarded term. The time and space derivatives have been discretized using the θ-method and a compact fourth-order finite difference method on a Shishkin mesh, respectively. Parameter uniform error estimates have been calculated in the $ L_\infty $ L ∞ norm. Some numerical examples have been considered to corroborate the theoretical results and compare the numerical results with existing methods in the literature. It is shown that the present approach provides improved results till date for the problem considered in this paper. [ABSTRACT FROM AUTHOR]

In this article, we explore the discontinuous Galerkin finite element method for two-parametric singularly perturbed convection-diffusion problems with a discontinuous source term. Due to the discontinuity in the source term, the problem typically shows a weak interior layer. Also, the presence of multiple perturbation parameters in the problem causes boundary layers on both sides of the boundary. In this work, we develop the nonsymmetric discontinuous Galerkin finite element method with interior penalties to handle the layer phenomenon. With the use of a typical Shishkin mesh, the domain is discretized, and a uniform error estimate is obtained. Numerical experiments are conducted to validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

This paper deals with an ε -uniform scheme for delay parabolic singularly perturbed problem (DPSPP). The considered problem is a convection–diffusion (C–D) type with the convection coefficient vanishing inside the domain. The turning point leads to the formation of two exponential boundary layers in the exact solution. We numerically solve the problem using the fitted mesh method and analyze the upwind scheme on a non-uniform mesh. We state some analytical results on the exact solution, which will be required in the convergence analysis of the proposed method. The proposed scheme has order of convergence almost one in space variable and one in time variable. The numerical findings practically support the theoretical results. [ABSTRACT FROM AUTHOR]

Some popular stabilization techniques, such as nonsymmetric interior penalty Galerkin (NIPG) method, have important application value in computational fluid dynamics. In this paper, we analyze a NIPG method on Shishkin mesh for a singularly perturbed convection diffusion problem, which is a typical simplified fluid model. According to the characteristics of the solution, the mesh and the numerical scheme, a new interpolation is designed for convergence analysis. More specifically, Gauß Lobatto interpolation and Gauß Radau interpolation are introduced inside and outside the layer, respectively. On the basis of that, by selecting special penalty parameters at different mesh points, we establish supercloseness of almost k + 1 order in an energy norm. Here k ≥ 1 is the degree of piecewise polynomials. Then, a simple post-processing operator is constructed, and it is proved that the corresponding post-processing can make the numerical solution achieve higher accuracy. In this process, a new analysis is proposed for the stability analysis of this operator. Finally, superconvergence is derived under a discrete energy norm. These conclusions can be verified numerically. Furthermore, numerical experiments show that the increase of polynomial degree k and mesh parameter N, the decrease of perturbation parameter ε or the use of over-penalty technology may increase the condition number of linear system. Therefore, we need to cautiously consider the application of high-order algorithm. [ABSTRACT FROM AUTHOR]

A reaction–diffusion problem with discontinuous source term and Dirichlet's boundary conditions on the unit square is considered in this paper. The proposed problem has been discretized using a combination of standard Galerkin finite element method (FEM) and non-symmetric discontinuous Galerkin finite element method with an interior penalty (NIPG) with bilinear elements. Layer adapted mesh of Shishkin type has been utilized to discretize the domain. Standard Galerkin FEM is applied on the layer part of the domain where the domain is dense enough and NIPG is applied to the outside layer part. By means of special choice of discontinuity-penalization parameters, the scheme is proved to be uniformly convergent of order $ \mathcal {O}(\varepsilon ^{1/4}N^{-1} + N^{-1} \ln N) $ O (ε 1 / 4 N − 1 + N − 1 ln ⁡ N). Numerical tests are carried out in support of theoretical findings. [ABSTRACT FROM AUTHOR]

This paper presents a numerical method for a class of singularly perturbed parabolic partial differential equations with integral boundary conditions (IBC). The solution to the considered problem exhibits pronounced boundary layers on both the left and right sides of the spatial domain. To address this challenging problem, we propose the use of the implicit Euler method for time discretization and a finite difference method on a well-designed piecewise uniform Shishkin mesh for spatial discretization. The integral boundary condition is approximated using Simpson's 13rule. The presented method demonstrates almost second-order uniform convergence in the discretization of the spatial derivative and first-order convergence in the discretization of the time derivative. To validate the applicability and accuracy of the proposed method, two illustrative examples are employed. The computational results not only accurately reflect the theoretical estimations but also highlight the method's effectiveness in capturing the intricate features of singularly perturbed parabolic partial differential equations with integral boundary conditions. [ABSTRACT FROM AUTHOR]

A class of two-parameter singularly perturbed nonlinear second order ordinary differential equations is considered in this article. A fitted mesh method which is a combination of finite difference scheme and a Shishkin mesh is developed to solve the problems. The method is proved to be essentially first order parameter independent convergent. Numerical experiments support the established theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a numerical scheme for a class of singularly perturbed delay parabolic convection–diffusion problems having Dirichlet boundary conditions is proposed. When the perturbation parameter tends to zero, the solution to these problems exhibits a boundary layer at the right side of the domain. The classical methods fail to provide good accuracy for these problems. To overcome this drawback, we proposed the Crank–Nicolson finite difference scheme to discretize in temporal direction on a uniform mesh and a nonstandard midpoint upwind scheme in spatial direction on a piecewise uniform Shishkin mesh. The resulting scheme has been shown to be uniformly stable and convergent with respect to perturbation parameter and mesh size. The scheme converges uniformly with the order of convergence OΔt2+N−1lnN2, where Δt is mesh size in temporal discretization and N is number of mesh elements in spatial discretization. To validate theoretical results, numerical experiments have been carried out and presented in the form of tables and graphs.

Fitted finite element methods are constructed for a singularly perturbed convection-diffusion problem in two space dimensions. Exponential splines as basis functions are combined with Shishkin meshes to obtain a stable parameter-uniform numerical method. These schemes satisfy a discrete maximum principle. If the diffusivity parameter is bounded below by some fixed positive constant, the numerical approximations converge, in L ∞ , at a rate of second order. Moreover, the numerical approximations converge at a rate of first order for all values of the singular perturbation parameter. [ABSTRACT FROM AUTHOR]

This article deals with the singularly perturbed nonlinear Volterra-Fredholm integrodifferential equations. Firstly, some priori bounds are presented. Then, the finite difference scheme is constructed on non-uniform mesh by using interpolating quadrature rules [5] and composite numerical integration formulas. The error estimates are derived in the discrete maximum norm. Finally, theoretical results are performed on two examples and they are compared for both Bakhvalov (B-type) and Shishkin (S-type) meshes. [ABSTRACT FROM AUTHOR]

In this article, we investigate a hybrid difference scheme for finding the numerical solution of a singularly perturbed second-order reaction-diffusion problem with a discontinuous source term. Such types of problems arise in the modeling of semiconductor devices and geophysical fluid dynamics etc. Solutions of these types of problems are difficult to obtain due to the presence of boundary and interior layers. A hybrid difference scheme i.e., cubic spline method and central finite difference approach, are applied on a fine region and coarse region, respectively. Shishkin mesh is utilized to generate the mesh point for the given domain. We use a second-order hybrid difference operator at the point of discontinuity. The solution rapidly changes in the interior layers and boundary layer. Truncation error is studied, and the stability of the method is analyzed. The proposed method is implemented on two problems, and numerical results are compared with the existing method, which shows that the proposed method is efficient for reducing maximum absolute errors and increasing the rate of convergence. [ABSTRACT FROM AUTHOR]

As a popular stabilization technique, the nonsymmetric interior penalty Galerkin (NIPG) method has significant application value in computational fluid dynamics. In this paper, we study the NIPG method for a typical two-dimensional singularly perturbed convection diffusion problem on a Shishkin mesh. According to the characteristics of the solution, the mesh and numerical scheme, a new composite interpolant is introduced. In fact, this interpolant is composed of a vertices-edges-element interpolant within the layer and a local L 2 -projection outside the layer. On the basis of that, by selecting penalty parameters on different types of interelement edges, we further obtain supercloseness of almost k + 1 2 order in an energy norm. Here k is the degree of piecewise polynomials. Numerical tests support our theoretical conclusion. [ABSTRACT FROM AUTHOR]

In this study, we consider singularly perturbed large negative shift parabolic reaction-diffusion with integral boundary condition. The continuous solution's properties are discussed. On a non-uniform Shishkin mesh, the spatial derivative is discretized using the tension spline method, and the temporal derivative is discretized using the Crank-Nicolson method. In order to handle the integral boundary condition, Simpson's rule is used. After conducting an error analysis, it was determined that the method was uniformly convergent. To support the theoretical findings, numerical examples are taken into account and solved for different values of the perturbation parameter and mesh sizes. It is proved to be uniformly convergent to almost second order. [ABSTRACT FROM AUTHOR]

In this article, a time delay parabolic convection-reaction-diffusion singularly perturbed problem with two small parameters is considered. We investigate the layer behavior of the solution for both smooth and non-smooth data. A numerical method to solve the problems described is developed using the Crank-Nicolson scheme to discretize the time-variable on a uniform mesh while a hybrid finite difference is applied for the space-variable. The hybrid scheme is a combination of the central, upwind and mid-point differencing on a piecewise uniform mesh of Shishkin type. The convergence analysis shows that the proposed method is uniformly convergent of second order in both space and time. Numerical experiments conducted on some test examples confirm the theoretical results. [ABSTRACT FROM AUTHOR]

This work presents a numerical solution to singularly perturbed Robin-type parabolic convection–diffusion problems. A hybrid method that combines the central difference scheme in the inner region and the midpoint of the upwind scheme in the outer region on a Shishkin mesh is utilized in the space direction, and an implicit Euler technique is used in the time direction. The analysis of convergence demonstrates that the present method converges with O(N−2ln2N+Δt). The theoretical findings are supported by two numerical examples. The upwind method proposed in the literature is improved by the present method.

In this study, we consider singularly perturbed large negative shift parabolic reaction–diffusion with integral boundary condition. The continuous solution's properties are discussed. On a non-uniform Shishkin mesh, the spatial derivative is discretized using the tension spline method, and the temporal derivative is discretized using the Crank–Nicolson method. In order to handle the integral boundary condition, Simpson's rule is used. After conducting an error analysis, it was determined that the method was uniformly convergent. To support the theoretical findings, numerical examples are taken into account and solved for different values of the perturbation parameter and mesh sizes. It is proved to be uniformly convergent to almost second order.

In this paper, a quadratic B-spline collocation method is developed to solve a singularly perturbed semilinear reaction–diffusion problem with a discontinuous source term. The discontinuous source term leads to a jump in the second-order derivative of the exact solution at the discontinuous point. A quadratic B-spline collocation method on a Shishkin-type mesh is used to discretized the singularly perturbed problem on the left and right sides of the discontinuous point, respectively. The collocation equations at the discontinuous point are obtained using the conditions satisfied at the discontinuous point. It is shown that the scheme is stable and almost second-order uniformly convergent. Numerical experiments support the theoretical results. [ABSTRACT FROM AUTHOR]

The goal of this study is to introduce a second-order computational method to solve Volterra-Fredholm integro-differential equations involving boundary layers. To solve numerically, we establish a finite difference scheme on the Shishkin mesh using a composite trapezoidal formulae for the integral part and interpolating quadrature rules and the linear exponential basis functions for the differential part. We establish that the numerical scheme and rate of convergence are both second-order and converge uniformly with reference to the small ε -parameter. The efficacy of the approach is supported by testing the numerical scheme's performance. [ABSTRACT FROM AUTHOR]

With the continuous development of science and technology, singularly perturbed problems have attracted more and more attentions from computational fluid scholars. In this paper, we discuss a singularly perturbed convection–diffusion problem with a discontinuous convection. Generally speaking, due to this discontinuity, an interior layer appears in the solution. A streamline diffusion finite element method on Shishkin mesh is used to solve this problem, and the optimal order of convergence in a modified streamline diffusion norm is derived. Numerical results are presented to support the theoretical conclusion. [ABSTRACT FROM AUTHOR]

A singularly perturbed reaction–diffusion problem posed on the unit square in R 2 is considered. To solve this problem numerically, a finite volume method (FVM) whose primal mesh is Shishkin is constructed; the FVM solution is piecewise bilinear on this mesh. Working in the standard energy norm, a superclose result (for the difference between the FVM solution and the Lagrange interpolant of the exact solution) is derived. This result yields an improved bound for the L 2 error of the FVM solution, and implies that a simple postprocessing of the FVM solution produces (in the energy norm) a higher-order approximation of the true solution. Next, we analyse errors in a balanced norm that is stronger than the energy norm; using a more complicated approximant of the exact solution from our piecewise bilinear space, we prove an optimal-order error bound and an associated supercloseness result showing that the difference between the FVM solution and our approximant is of higher order than the error itself. Finally, numerical experiments demonstrate the sharpness of our error bounds. [ABSTRACT FROM AUTHOR]

We have considered a third order singularly perturbed reaction diffusion equations with integral boundary condition. On a Shishkin mesh, a numerical technique based on a finite difference scheme is provided. The proposed approach is almost first order convergent. In the discrete norm, an error estimate is obtained. Numerical examples are provided to validate the theoretical estimates. [ABSTRACT FROM AUTHOR]

In this article, we analyzed a general system of first order singularly perturbed semi-linear equations with distinct perturbation parameters in the unit interval. As boundary layers are expected near the origin in the solution components, variants of piecewise uniform meshes, introduced by Shishkin, are constructed to discretize the unit interval and standard finite difference scheme is used to discretize the equations. Parameter uniform convergence of the composed numerical method is proved. A continuation method is used to compute the numerical solution of the non-linear problem and numerical illustrations are given in support. [ABSTRACT FROM AUTHOR]

This paper presents a non-symmetric interior penalty Galerkin (NIPG) finite element scheme for one-dimensional singularly perturbed reaction-diffusion equations with discontinuous diffusion. Such problems often arise in the modelling of phase transitions. The solution of the considered class of problem is known to exhibit boundary and interior layers. Priori bounds are given for the solution and its derivatives and NIPG finite element scheme is constructed for the numerical approximation of the given problem. Error estimates for the proposed scheme are derived in the energy norm as well as the balanced norm. Error bounds have been proved to be uniform with respect to the perturbation parameter ε . Numerical experiments are conducted in support of the theoretical estimates. [ABSTRACT FROM AUTHOR]

This article focuses on a class of fourth-order singularly perturbed convection diffusion equations (SPCDE) with integral boundary conditions (IBC). A numerical method based on a finite difference scheme using Shishkin mesh is presented. The proposed method is close to the first-order convergent. The discrete norm yields an error estimate and theoretical estimations are tested by numerical experiments.

In this study, a robust higher-order numerical method for solving singularly perturbed parabolic reaction-diffusion problems is presented. The Crank-Nicolson method is applied to discretize the time derivative on a uniform mesh. On a Shishkin mesh, the space derivative is discretized using a hybrid numerical method that combines the cubic spline in tension method for the boundary layer regions with the central difference method for the outer region. Theoretically, we proved that the proposed hybrid numerical method is second-order in the outer region and fourth-order in the boundary layer regions in the space direction. As a result of this, the proposed method produces an almost second-order rate of convergence in the time domain and a higher-order rate of convergence in the space domain. The newly developed method is numerically demonstrated to be uniformly convergent at higher-order, independent of the perturbation parameter. Three numerical examples are computed to support the theoretical results.