Shishkin Mesh Research Articles

Convection dominated singularly perturbed problems on a metric graph

This article concentrates on singularly perturbed static convection–diffusion equations with varying coefficients on a metric graph G=(V,E). Our interest is in the convection dominated situation which is described by a small parameter ϵ>0 in front of the diffusion term. As ϵ→0, the reduced problem may exhibit boundary layers at the multiple vertices as well as at the simple nodes. We analyze the possible scenarios and validate the results in several test cases. We investigate several exemplary graphs and use an upwind finite difference method on a piece-wise Shishkin mesh. Error estimates are also discussed to show ϵ-uniform convergence.

Fitted mesh method for singularly perturbed fourth order differential equation of convection diffusion type with integral boundary condition

<abstract><p>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.</p></abstract>

A parameter-uniform weak Galerkin finite element method for a coupled system of singularly perturbed reaction-diffusion equations

The aim of this paper to investigate a weak Galerkin finite element method (WG-FEM) for solving a system of coupled singularly perturbed reaction-diffusion equations. Each equation in the system has perturbation parameter of different magnitude and thus, the solutions will exhibit two distinct but overlapping boundary layers near each boundary of the domain. The proposed method is applied to the coupled system on Shishkin mesh to solve the problem theoretically and numerically. Elimination of the interior unknowns efficiently from the discrete solution system reduces the degrees of freedom and, thus the number of unknown in the discrete solution is comparable with the standard finite element scheme. The stability and error analysis of the proposed method on the Shishkin mesh are presented. We show that the method convergences of order O(N?k lnk N) in the energy norm, uniformly with respect to the perturbation parameter. Moreover, the optimal convergence rate of O(N?(k+1)) in the L2-norm and the superconvergence rate of O((N?2k ln2k N) in the discrete L?-norm is observed numerically. Finally, some numerical experiments are carried out to verify numerically theory.

A Numerical Approach for Diffusion‐Dominant Two‐Parameter Singularly Perturbed Delay Parabolic Differential Equations

A numerical scheme is developed to solve a large time delay two‐parameter singularly perturbed one‐dimensional parabolic problem in a rectangular domain. Two small parameters multiply the convective and diffusive terms, which determine the width of the left and right lateral surface boundary layers. Uniform mesh and piece‐wise uniform Shishkin mesh discretization are applied in time and spatial dimensions, respectively. The numerical scheme is formulated by using the Crank–Nicolson method on two consecutive time steps and the average central finite difference approximates in spatial derivatives. It is proved that the method is uniformly convergent, independent of the perturbation parameters, where the convection term is dominated by the diffusion term. The resulting scheme is almost second‐order convergent in the spatial direction and second‐order convergent in the temporal direction. Numerical experiments illustrate theoretical findings, and the method provides more accurate numerical solutions than prior literature.

A parameter uniform method for two-parameter singularly perturbed boundary value problems with discontinuous data

We consider two-parameter singularly perturbed problems of reaction-convection-diffusion type in one dimension. The convection coefficient and source term are discontinuous at a point in the domain. The problem is numerically solved using the upwind difference method on an appropriately defined Shishkin-Bakhvalov mesh. At the point of discontinuity, a three-point difference scheme is used. A convergence analysis is given and the method is shown to be first-order uniformly convergent with respect to the perturbation parameters. The numerical results presented in the paper confirm our theoretical results of first-order convergence. Summing up:• The Shishkin-Bakhvalov mesh is graded in the layer region and uniform in the outer region as shown in the graphical abstract.• The method presented here has uniform convergence of order one in the supremum norm.• The numerical orders of convergence obtained in numerical examples with Shishkin- Bakhvalov mesh are better than those for Shishkin mesh.

Error estimations of a weak Galerkin finite element method for a linear system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion equations in the energy and balanced norms

<abstract><p>This paper introduces a weak Galerkin finite element method for a system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion problems. The proposed method is independent of parameter and uses piecewise discontinuous polynomials on interior of each element and constant on the boundary of each element. By the Schur complement technique, the interior unknowns can be locally efficiently eliminated from the resulting linear system, and the degrees of freedom of the proposed method are comparable with the classical FEM. It has been reported that the energy norm is not adequate for singularly perturbed reaction-diffusion problems since it can not efficiently reflect the behaviour of the boundary layer parts when the diffusion coefficient is very small. For the first time, the error estimates in the balanced norm has been presented for a system of coupled singularly perturbed problems when each equation has different parameter. Optimal and uniform error estimates have been established in the energy and balanced norm on an uniform Shishkin mesh. Finally, we carry out various numerical experiments to verify the theoretical findings.</p></abstract>

Numerical Scheme for Singularly Perturbed Mixed Delay Differential Equation on Shishkin Type Meshes

Two non-uniform meshes used as part of the finite difference method to resolve singularly perturbed mixed-delay differential equations are studied in this article. The second-order derivative is multiplied by a small parameter which gives rise to boundary layers at x=0 and x=3 and strong interior layers at x=1 and x=2 due to the delay terms. We prove that this method is almost first-order convergent on Shishkin mesh and is first-order convergent on Bakhvalov–Shishkin mesh. Error estimates are derived in the discrete maximum norm. Some examples are provided to validate the theoretical result.

Boundary Layer Resolving Exact Difference Scheme for Solving Singularly Perturbed Convection-Diffusion-Reaction Equation

This paper considers the numerical treatment of singularly perturbed time-dependent convection-diffusion-reaction equation. The diffusion term of the equation is multiplied by a small perturbation parameter (ε), which takes an arbitrary value in the interval (0, 1]. For small values of ε, the solution of the equation exhibits an exponential boundary layer which makes it difficult to solve it analytically or using classical numerical methods. We proposed numerical schemes using the Crank–Nicolson method in time derivative discretization and the nonstandard finite difference method (exact finite difference method) in space derivative discretization on a uniform and piecewise uniform Shishkin mesh. The existence of unique discrete solutions and the stability of the schemes are discussed and proved. Uniform convergence of the schemes is proved. The formulated schemes converge uniformly with linear order of convergence. The method on Shishkin mesh possesses boundary layer resolving property. We validated the methods by considering two numerical examples for different values of ε and mesh length.

Layer resolving numerical scheme for singularly perturbed parabolic convection-diffusion problem with an interior layer

Singularly perturbed parabolic convection-diffusion problem with interior layer is a type of singularly perturbed boundary value problems which have sign change properties in the coefficient function of the convection term. This paper introduces a layer resolving numerical scheme for solving the numerical solution of the singularly perturbed parabolic convection-diffusion problem exhibiting interior layer due to the convection coefficient. The scheme is formulated by discretizing the temporal variable on uniform mesh and discretize the spatial one on piecewise uniform mesh of the Shishkin mesh type. The resulting scheme is shown to be almost first order convergent. Theoretical investigations are confirmed by numerical experiments. Moreover, the present scheme is:•Stable,•Consistent and•Gives more accurate solution than existing methods in the literature.

A fitted numerical method for a singularly perturbed Fredholm integro-differential equation with discontinuous source term

In this article, a singularly perturbed second-order Fredholm integro-differential equation with a discontinuous source term is examined. An exponentially-fitted numerical method on a Shishkin mesh is applied to solve the problem. The method is shown to be uniformly convergent with respect to the singular perturbation parameter. Some numerical results are given, which validate the theoretical results.

Error analysis of a weak Galerkin finite element method for two-parameter singularly perturbed differential equations in the energy and balanced norms

A weak Galerkin finite element method is proposed for solving singularly perturbed problems with two parameters. A robust uniform optimal convergence has been proved in the corresponding energy and a stronger balanced norms using piecewise higher order discontinuous polynomials on a piecewise uniform Shishkin mesh. Finally, we give some numerical experiments to support theoretical results.

A computational method for a two-parameter singularly perturbed elliptic problem with boundary and interior layers

In this article, we investigate a two-dimensional (2-D) singularly perturbed convection–reaction–diffusion elliptic type problem where two parameters ϵ and μ multiply the diffusion and convection terms, respectively. Furthermore, we assume that jump discontinuities exist in the source term along the x- and y-axis. Due to the presence of perturbation parameters, the solutions to such problems show boundary and corner layers. Moreover, the discontinuity in the source term adds the interior layers to the solution whose suitable numerical approach is the important goal of this article. A numerical approach is carried out using an upwind finite-difference technique that includes an appropriate layer-adapted piecewise uniform Shishkin mesh. Some examples are presented which show the good performance of the proposed method and the agreement with the theoretical analysis.

NIPG Method on Shishkin Mesh for Singularly Perturbed Convection–Diffusion Problem with Discontinuous Source Term

In this paper, we provide numerical approaches for singularly perturbed convection–diffusion problem with a discontinuous source term. As a result of the discontinuity in the source term, the problem highlights an exponential-type boundary layer and a weak interior layer. To handle both the layers at the same time, we design a nonsymmetric discontinuous Galerkin finite element technique with interior penalties (NIPG). The problem is solved using the standard Shishkin mesh. In the DG norm, uniform error estimates are derived for the problem. To verify the theoretical estimates, numerical tests are carried out.

A parameter uniform higher order scheme for 2D singularly perturbed parabolic convection–diffusion problem with turning point

In this article, we construct and analyze a higher order numerical method for a class of two dimensional parabolic singularly perturbed problem (PSPP) of convection–diffusion (C–D) type for the case when the convection coefficient is vanishing inside the domain. The asymptotic behavior of the exact solution is studied for the considered problem. Peaceman–Rachford scheme on a uniform mesh is used for time discretization and a hybrid scheme on the Bakhvalov–Shishkin mesh is applied for spatial discretization. The convergence analysis shows that the proposed scheme is uniformly convergent with respect to parameter ɛ. It is established that the hybrid scheme on the Bakhvalov–Shishkin mesh has second order of convergence despite the use of the standard Shishkin mesh which leads to order reduction due to the presence of a logarithmic term. The numerical results corroborate the theoretical expectations and show high accuracy of the proposed scheme over the hybrid scheme on a standard Shishkin mesh. Also, the hybrid scheme is compared with the upwind scheme through the numerical results.

Local discontinuous Galerkin method for a third‐order singularly perturbed problem of reaction–diffusion type

AbstractA third‐order singularly perturbed problem of reaction–diffusion type is solved numerically by the local discontinuous Galerkin (LDG) method. By using the local Gauss–Radau projection for the primal variable u, but local L2 projection for the two auxiliary variables p and q, we obtain an optimal convergence rate of order (or up to a logarithmic factor) in the energy norm, uniformly in the singular perturbation parameter. Here is the degree of the piecewise polynomials used in the discrete space and N is the number of mesh intervals. The layer‐adapted meshes are used including a standard Shishkin mesh, a Bakhvalov–Shishkin mesh, and a Bakhvalov‐type mesh. Numerical experiments are given to show the sharpness of our theoretical results.

A splitting uniformly convergent method for one-dimensional parabolic singularly perturbed convection-diffusion systems

In this paper we deal with solving robustly and efficiently one-dimensional linear parabolic singularly perturbed systems of convection-diffusion type, where the diffusion parameters can be different at each equation and even they can have different orders of magnitude. The numerical algorithm combines the classical upwind finite difference scheme to discretize in space and the fractional implicit Euler method together with an appropriate splitting by components to discretize in time. We prove that if the spatial discretization is defined on an adequate piecewise uniform Shishkin mesh, the fully discrete scheme is uniformly convergent of first order in time and of almost first order in space. The technique used to discretize in time produces only tridiagonal linear systems to be solved at each time level; thus, from the computational cost point of view, the method we propose is more efficient than other numerical algorithms which have been used for these problems. Numerical results for several test problems are shown, which corroborate in practice both the uniform convergence and the efficiency of the algorithm.

Alternating direction implicit method for singularly perturbed 2D parabolic convection–diffusion–reaction problem with two small parameters

ABSTRACT In this article, we construct and analyse an Alternating Direction Implicit (ADI) scheme for singularly perturbed 2D parabolic convection–diffusion–reaction problems with two small parameters. We consider the operator-splitting ADI finite difference scheme for time stepping on a uniform mesh and a simple upwind-difference scheme for spatial discretization on a specially designed piecewise-uniform Shishkin mesh. The resulting scheme is proved to be uniformly convergent of order , where N, M are the spatial and temporal parameters respectively. Numerical experiments confirm the theoretical results and the effectiveness of the proposed method.

Shishkin mesh based septic Hermite interpolation algorithm for time-dependent singularly perturbed convection–diffusion models

Shishkin mesh based septic Hermite interpolation algorithm for time-dependent singularly perturbed convection–diffusion models

A computational approach for a two-parameter singularly perturbed system of partial differential equations with discontinuous coefficients

This work aims at obtaining a numerical approximation to the solution of a two-parameter singularly perturbed convection-diffusion-reaction system of partial differential equations with discontinuous coefficients. This discontinuity, together with small values of the perturbation parameters, causes interior and boundary layers to appear in the solution. To obtain appropriate point-wise accuracy, we have considered a central finite-difference approach in the space variable which is defined on a piecewise uniform Shishkin mesh and an implicit Euler scheme in the temporal variable defined on a uniform mesh. Some computational experiments have been performed, which confirm the theoretical findings.

Analysis of a Fitted Finite Difference Scheme for a Semilinear System of Singularly Perturbed Reaction-Diffusion Equations Having Discontinuous Source Term

In this paper, a system of an arbitrary number of singularly perturbed semilinear differential equations of reaction-diffusion type having discontinuous source term is examined, for the case in which the diffusion parameters associated with each equation of the considered system are assumed to be different in magnitude. In addition to the occurrence of the boundary layers, the solution exhibits the overlapping and interacting internal layers due to the existence of discontinuity in the data. A central finite difference scheme is used in conjunction with a suitable piecewise uniform Shishkin mesh and a Bakhvalov mesh, to construct the numerical method. Using discrete Green’s function technique, the proposed numerical scheme has been proved to be an almost second-order of uniform convergent for the Shishkin mesh and second-order of uniform convergent for the Bakhvalov mesh, independent of the perturbation parameters. Numerical test examples are presented to demonstrate the performance of the numerical scheme.