Perturbed Convection Diffusion Research Articles

Convergence and supercloseness of a finite element method for a singularly perturbed convection–diffusion problem on an L-shaped domain

Convergence and supercloseness of a finite element method for a singularly perturbed convection–diffusion problem on an L-shaped domain

Higher order uniformly convergent NIPG methods for 1-d singularly perturbed problems of convection–diffusion type

In this paper, a higher order NIPG method on a S-type meshes has been developed and analyzed for the singularly perturbed convection–diffusion problems. We prove that the method is uniformly convergent with order k in ε-weighted DG energy norm, where k is the degree of piecewise polynomial in finite element space. Numerical experiments support these theoretical results. Moreover, the numerical results show that the NIPG method has a supercloseness property of order k+1 in associated norm if k is odd. But there is no supercloseness property if k is even.

Uniformly convergent hybrid schemes for solutions and derivatives in quasilinear singularly perturbed BVPs

In this paper, a class of hybrid difference schemes with variable weights on Bakhvalov–Shishkin mesh is proposed to compute both the solution and the derivative in quasilinear singularly perturbed convection–diffusion boundary value problems. The parameter-uniform second-order convergence of approximating to the solution and the derivative on Bakhvalov–Shishkin mesh and that of nearly second-order on Shishkin mesh are proved clearly by use of an (l∞,l1)-stability property, where the former sufficient conditions for uniform convergence are modestly relaxed on Bakhvalov–Shishkin mesh and are clarified on Shishkin mesh. The numerical examples support the proposed schemes with new sufficient conditions and their error estimates.

A Numerical Method for Singularly Perturbed Second Order Coupled System of Convection–Diffusion Robin Type Boundary Value Problems with Discontinuous Source Term

In this paper, a numerical method for a weakly coupled system of singularly perturbed convection–diffusion second order ordinary differential equations with discontinuous source term subject to Robin type boundary conditions is presented. A fitted mesh method has been used to obtain the difference scheme to solve the system of equations on a piecewise uniform Shishkin mesh. An error estimate is derived to show that the method is uniformly convergent with respect to singular perturbation parameter. Numerical results are provided to illustrate the theoretical results.

An adaptive mesh strategy for singularly perturbed convection diffusion problems

In this paper, a new adaptive mesh strategy has been developed for solving convection dominated, convection–diffusion singularly perturbed problems (SPP) using second order central difference schemes. Our strategy uses a novel, entropy-like variable as the adaptation parameter for convection diffusion SPP. Further, unlike the popular layer adapted meshes mainly by Bakhvalov (B-type) and Shishkin (S-type), no pre-knowledge of the location and width of the layers (boundary as well as interior) is needed. The method is completely free of arbitrary perturbation parameters ∊ (robust) and results in oscillation free solutions to a range of convection–diffusion SPP. Numerical results for various test problems: linear (boundary layers with and without turning points), nonlinear and systems of coupled equations are presented. This method is expected to aid in robust computations of convection dominated SPP.

A uniformly convergent nonstandard finite difference scheme for a system of convection–diffusion equations

We propose a nonstandard finite difference method to integrate coupled systems of singularly perturbed convection–diffusion equations. We prove that the underlying discrete operator satisfies a stability property in the maximum norm. This fact is used to prove that the proposed method converges uniformly with respect to the singular perturbation parameters and the convergence rate is linear with respect to the step size. Numerical investigations are presented to confirm our theoretical findings. We also show that the proposed approach compares well with existing methods.

A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems

A finite difference method for a time-dependent singularly perturbed convection–diffusion–reaction problem involving two small parameters in one space dimension is considered. We use the classical implicit Euler method for time discretization and upwind scheme on the Shishkin–Bakhvalov mesh for spatial discretization. The method is analysed for convergence and is shown to be uniform with respect to both the perturbation parameters. The use of the Shishkin–Bakhvalov mesh gives first-order convergence unlike the Shishkin mesh where convergence is deteriorated due to the presence of a logarithmic factor. Numerical results are presented to validate the theoretical estimates obtained.

Robust numerical scheme for singularly perturbed convection–diffusion parabolic initial–boundary-value problems on equidistributed grids

In this article, we study the numerical solution of singularly perturbed parabolic convection–diffusion problems exhibiting regular boundary layers. To solve these problems, we use the classical upwind finite difference scheme on layer-adapted nonuniform meshes. The nonuniform meshes are obtained by equidistributing a positive monitor function, which depends on the second-order spatial derivative of the singular component of the solution. The truncation error and the stability analysis are obtained. Parameter-uniform error estimates are derived for the numerical solution. Semilinear IBVPs are also solved. Numerical experiments are carried out to support the theoretical results.

Superconvergence using pointwise interpolation in convection–diffusion problems

Considering a singularly perturbed convection–diffusion problem, we present an analysis for a superconvergence result using pointwise interpolation of Gauß–Lobatto type for higher-order streamline diffusion FEM. We show a useful connection between two different types of interpolation, namely a vertex–edge–cell interpolant and a pointwise interpolant. Moreover, different postprocessing operators are analysed and applied to model problems.

A robust adaptive method for singularly perturbed convection–diffusion problem with two small parameters

Numerical treatment for a one dimensional two-parameter singularly perturbed convection–diffusion equation is proposed and analyzed. The upwind finite difference method is applied to the problem. Maximum-norm a posteriori error estimates are obtained and used to establish an adaptive algorithm. The uniform convergence of the new algorithm is proved under feasible conditions of two parameters. Numerical experiments are given to confirm the analytical results.

Data perturbation stability of difference schemes on uniform grids for a singularly perturbed convection–diffusion equation

An approach to the study of the conditioning of difference schemes and their stability to data perturbations is developed for the Dirichlet problem for a singularly perturbed convection-diffusion ordinary differential equation with the perturbation parameter e, e 2 (0;1]. We consider a standard difference scheme, which is a monotone scheme on a uniform grid, and a special scheme of the grid solution decomposition method. Constructing the special scheme, we use a decomposition of the grid solution into a regular and a singular components that are the solutions to grid subproblems considered on uniform grids. The following facts are proved for the standard scheme: (a) the scheme converges only for N-1 = o(ε) at the rate (N-1 (ε +N-1)-1), , where N +1 is the number of grid nodes, (b) the scheme is not e-uniformly well conditioned and stable to perturbations, (c) the condition number of the scheme has the order (ε-1 δ-2, where δ is the accuracy of the grid solution, δ = δst ≈ N-1 (ε +N-1)-1. In the case of convergence of the standard scheme proved theoretically, the actual accuracy of the computed solution decreases with the decrease in the parameter e under the presence of perturbations and may completely vanish for a sufficiently small e (namely, under the condition t = (lnε-1 +lnδ-2st), where t is the number of digits in the machine word). At the same time, the special scheme of the solution decomposition method converges e-uniformly in the uniform norm at the rate (N-1 lnN); in the variables e and d, the special scheme is e-uniformly well conditioned and stable to grid problem data perturbations; the condition number of the scheme of the decomposition method is of the order (δ-2 lnδ-1), δ = δdec ≈ N-1 lnN.

PSEUDO-MULTI-SCALE FUNCTIONS FOR THE STABILIZATION OF CONVECTION-DIFFUSION EQUATIONS ON RECTANGULAR GRIDS

We propose a finite element method of Petrov-Galerkin type for a singularly perturbed convection diffusion problem on a discretization consisting of rectangular elements. The method is based on enriching the finite-element space with a combination of multiscale and residual-free bubble functions. These functions require the solution of the original differential problem, which makes the method quite expensive, especially in two dimensions. Therefore, we instead employ their cheap, yet efficient approximations, using only a few nodes in each element. Several numerical tests confirm the good performance of the corresponding numerical method.

An Initial Value Technique for Singularly Perturbed Convection–Diffusion Problems with a Negative Shift

In this paper, a numerical method named as Initial Value Technique (IVT) is suggested to solve the singularly perturbed boundary value problem for the second order ordinary differential equations of convection–diffusion type with a delay (negative shift). In this technique, the original problem of solving the second order equation is reduced to solving two first order differential equations, one of which is singularly perturbed without delay and other one is regular with a delay term. The singularly perturbed problem is solved by the second order hybrid finite difference scheme, whereas the delay problem is solved by the fourth order Runge–Kutta method with Hermite interpolation. An error estimate is derived by using the supremum norm. Numerical results are provided to illustrate the theoretical results.

Pointwise estimates of the SDFEM for convection–diffusion problems with characteristic layers

A model singularly perturbed convection–diffusion problem in two space dimensions is considered. The problem is solved by a streamline diffusion finite element method (SDFEM) that uses piecewise bilinear finite elements on a Shishkin mesh. We prove that the method is convergent, independently of the diffusion parameter ε, with a pointwise accuracy of almost order 7/4 away from the characteristic layers. Numerical experiments support these theoretical results.

Asymptotic Initial Value Technique for singularly perturbed convection–diffusion delay problems with boundary and weak interior layers

In this paper a numerical method named as Asymptotic Initial Value Technique (AIVT) is suggested to solve the singularly perturbed boundary value problem for the second order ordinary delay differential equation with the discontinuous convection–diffusion coefficient term. In this technique, the original problem of solving the second order differential equation is reduced to solving three first order differential equations, one of which is a delay differential equation and other two are singularly perturbed problems. The singularly perturbed problems are solved by the second order hybrid finite difference scheme, whereas the delay problem is solved by the fourth order Runge–Kutta method with Hermite interpolation. An error estimate is derived by using the supremum norm and it is of order O ( ε + N − 2 ln 2 N ) , where N and ε are the discretization parameter and the perturbation parameter, respectively. Numerical results are provided to illustrate the theoretical results.

Novel fitted operator finite difference methods for singularly perturbed elliptic convection–diffusion problems in two dimensions

We consider a class of singularly perturbed elliptic problems posed on a unit square. These problems are solved by using fitted mesh methods by many researchers but no attempts are made to solve them using fitted operator methods, except our recent work on reaction–diffusion problems [J.B. Munyakazi and K.C. Patidar, Higher order numerical methods for singularly perturbed elliptic problems, Neural Parallel Sci. Comput. 18(1) (2010), pp. 75–88]. In this paper, we design two fitted operator finite difference methods (FOFDMs) for singularly perturbed convection–diffusion problems which possess solutions with exponential and parabolic boundary layers, respectively. We observe that both of these FOFDMs are ϵ-uniformly convergent. This fact contradicts the claim about singularly perturbed convection–diffusion problems [Miller et al. Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996] that ‘when parabolic boundary layers are present, …, it is not possible to design an ϵ-uniform FOFDM if the mesh is restricted to being a uniform mesh’. We confirm our theoretical findings through computational investigations and also found that we obtain better results than those of Linß and Stynes [Appl. Numer. Math. 31 (1999), pp. 255–270].

Convergence analysis of the LDG method applied to singularly perturbed problems

AbstractConsidering a two‐dimensional singularly perturbed convection–diffusion problem with exponential boundary layers, we analyze the local discontinuous Galerkin (DG) method that uses piecewise bilinear polynomials on Shishkin mesh. A convergence rate O(N‐1 lnN) in a DG‐norm is established under the regularity assumptions, while the total number of mesh points is O(N2). The rate of convergence is uniformly valid with respect to the singular perturbation parameter ε. Numerical experiments indicate that the theoretical error estimate is sharp. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Convergence phenomena of \documentclass{article} \usepackage{amsmath,amsfonts}\pagestyle{empty}\begin{document} $\mathcal{Q}_p$ \end{document}‐elements for convection–diffusion problems

AbstractWe present a numerical study for singularly perturbed convection–diffusion problems using higher order Galerkin and streamline diffusion finite element method. We are especially interested in convergence and superconvergence properties with respect to different interpolation operators. For this we investigate pointwise interpolation and vertex‐edge‐cell interpolation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Pointwise error estimates of the bilinear SDFEM on Shishkin meshes

AbstractA model singularly perturbed convection–diffusion problem in two space dimensions is considered. The problem is solved by a streamline diffusion finite element method (SDFEM) that uses piecewise bilinear finite elements on a Shishkin mesh. We prove that the method is convergent, independently of the diffusion parameter ε, with a pointwise accuracy of almost order 11/8 outside and inside the boundary layers. Numerical experiments support these theoretical results. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

An efficient mixed asymptotic-numerical scheme for singularly perturbed convection diffusion problems

In this paper, we present and analyze a mixed asymptotic-numerical method for the solution of singularly perturbed convection diffusion problems. The technique, used in this work, is the careful factorization of original problem into two explicit problems which are independent of perturbation parameter contaminating the solution. The first problem, so obtained, is the degenerate equation which corresponds to the outer solution while the second problem is obtained by making use of suitable stretching transformation and it corresponds to the inner solution. The degenerate equation is solved numerically using q-stage Runge–Kutta method. The second problem, which retains the order of the original problem, is solved analytically. Possible extensions of the present technique to differential equations with delay as well as advance and to the nonlinear problems are also discussed. A comparative study of the present method with some state of art existing numerical schemes is carried out by means of several linear and nonlinear test examples that demonstrates the effectiveness and the potential of the present approach.