Method For Singular Perturbation Problems Research Articles

A Posteriori Error Analysis of Defect Correction Method for Singular Perturbation Problems With Discontinuous Coefficient and Point Source

Abstract The paper presents a defect correction method to solve singularly perturbed problems with discontinuous coefficient and point source. The method combines an inexpensive, lower-order stable, upwind difference scheme and a higher-order, less stable central difference scheme over a layer-adapted mesh. The mesh is designed so that most mesh points remain in the regions with rapid transitions. A posteriori error analysis is presented. The proposed numerical method is analyzed for consistency, stability, and convergence. The error estimates of the proposed numerical method satisfy parameter-uniform second-order convergence on the layer-adapted grid. The convergence obtained is optimal because it is free from any logarithmic term. The numerical analysis confirms the theoretical error analysis.

Superconvergence analysis of the conforming discontinuous Galerkin method on a Bakhvalov-type mesh for singularly perturbed reaction–diffusion equation

The conforming discontinuous Galerkin (CDG) method maximizes the utilization of all degrees of freedom of the discontinuous Pk polynomial to achieve a convergence rate two orders higher than its counterpart conforming finite element method employing continuous Pk element. Despite this superiority, there is little theory of the CDG methods for singular perturbation problems. In this paper, superconvergence of the CDG method is studied on a Bakhvalov-type mesh for a singularly perturbed reaction–diffusion problem. For this goal, a pre-existing least squares method has been utilized to ensure better approximation properties of the projection. On the basis of that, we derive superconvergence results for the CDG finite element solution in the energy norm and L2-norm and obtain uniform convergence of the CDG method for the first time.

Numerical analysis of singularly perturbed parabolic reaction diffusion differential difference equations

The authors present numerical analysis of singularly perturbed parabolic problems. The previous papers in this direction focussed on the convection-diffusion problems with regular type of boundary layers. It is very difficult and almost impossible as stated in the literature, e.g, in Chapter 14 of ‘Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions’ (1996) by Miller, John JH and O'Riordan, Eugene and Shishkin, Grigorii I, to capture singularities due to parabolic layers and develop a robust scheme for reaction-diffusion problems with general values of space shifts. The work is in progress now and this is the first paper in this direction. In this work, we are investigating such problems and develop a robust numerical scheme using Mickens's non-standard finite difference scheme and special mesh. Furthermore, numerical examples have been presented to verify the theoretical findings.

An Extended Finite Difference Method for Singular Perturbation Problems on a Non-Uniform Mesh

An extended second order finite difference method on a variable mesh is proposed for the solution of a singularly perturbed boundary value problem. A discrete equation is achieved on the non uniform mesh by extending the first and second order derivatives to the higher order finite differences. This equation is solved efficiently using a tridiagonal solver. The proposed method is analysed for convergence, and second order convergence is derived. Model examples are solved by the proposed scheme and compared with available methods in the literature to uphold the method.

Numerical solution of time‐fractional singularly perturbed convection–diffusion problems with a delay in time

This work is concerned with the development of a stable finite difference method (SFDM) for time‐fractional singularly perturbed convection–diffusion problems with a delay in time. The fractional derivative is considered in the Caputo sense. The SFDM is constructed based on the stability of the analytical solution. Unlike the other classical numerical methods for singular perturbation problems, this method works nicely on a uniform mesh. The method is easily adaptable on Shishkin mesh and also on any graded mesh in space. Error estimates are presented to show the convergence of the numerical scheme. To support the theory, numerical results are presented in tables for different values of the fractional derivative parameter and perturbation parameter.

Variable mesh non polynomial spline method for singular perturbation problems exhibiting twin layers

In this paper, we descend a variable mesh finite difference scheme based on non polynomial spline approximation for the solution of singular perturbation problems with twin boundary layers. We develop the discretization equation for the problem using the condition of continuity for the first order derivatives of the variable mesh non polynomial spline at the interior nodes. The discrete invariant imbedding algorithm is utilized to solve the tridiagonal system obtained by the method. Endeavor examples are illustrated and maximum absolute errors in comparison to the other methods in the literature are shown to vindicate the method.

Special Finite Difference Method for Singular Perturbation Problems with One-End Boundary Layer

In this paper, we have presented a special finite difference method for solving a singular perturbation problem with layer behaviour at one end. In this method, we have used a second order finite difference approximation for the second derivative, a modified second order upwind finite difference approximation for the first derivative and a second order average difference approximation for y to reduce the global error and retaining tridiagonal system. Then the discrete invariant imbedding algorithm is used to solve the tridiagonal system. This method controls the rapid changes that occur in the boundary layer region and it gives good results in both cases i.e., h ≤ ε and ε << h. The existence and uniqueness of the discrete problem along with stability estimates are discussed. Also we have discussed the convergence of the method. We have presented maximum absolute errors for the standard examples chosen from the literature.

Scattering by a highly oscillating surface

The boundary function method [A. B. Vasil'eva, V. F. Butuzov, and L. V. Kalachev, The boundary function method for singular perturbation problems, SIAM Studies in Applied Mathematics, Philadelphia, 1995] is used to build an asymptotic expansion at any order of accuracy of a scalar time‐harmonic wave scattered by a perfectly reflecting doubly periodic surface with oscillations at small and large scales. Error bounds are rigorously established, in particular in an optimal way on the relevant part of the field. It is also shown how the maximum principle can be used to design a homogenized surface whose reflected wave yields a first‐order approximation of the actual one. The theoretical derivations are illustrated by some numerical experiments, which in particular show that using the homogenized surface outperforms the usual approach consisting in setting an effective boundary condition on a flat boundary. Copyright © 2014 John Wiley & Sons, Ltd.

High order multiplication perturbation method for singular perturbation problems

This paper presents a high order multiplication perturbation method for singularly perturbed two-point boundary value problems with the boundary layer at one end. By the theory of singular perturbations, the singularly perturbed two-point boundary value problems are first transformed into the singularly perturbed initial value problems. With the variable coefficient dimensional expanding, the non-homogeneous ordinary differential equations (ODEs) are transformed into the homogeneous ODEs, which are then solved by the high order multiplication perturbation method. Some linear and nonlinear numerical examples show that the proposed method has high precision.

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].

Education

The Education section in this issue of SIAM Review features two articles with two unique flavors. The first brings together distinct topics first introduced in the undergraduate curriculum and is suitable for an undergraduate audience. The second is a brief renormalization group tutorial for a graduate curriculum. In “Modified Gershgorin Disks for Companion Matrices,” author Aaron Melman connects zeros of polynomials with the properties of their corresponding companion matrix. He begins with the basic notion that the zeros of a polynomial are contained within the union of Gershgorin disks of the corresponding companion matrix. While this is an elegant connection, the union of disks can be a fairly large set. The author points out that a careful examination of the structure of the companion matrix and some judicious manipulation can restrict these sets much further. The resulting discussion makes a nice module for an undergraduate course in linear algebra or analysis. This article also includes MATLAB code for rendering the restricted sets where the roots reside. The second article, “The Renormalization Group: A Perturbation Method for the Graduate Curriculum,” by Eleftherios Kirkinis, provides a gentle introduction to renormalization group (RG) methods for singular perturbation problems. The authors make the case that RG methods provide a more consistent approach to perturbation problems than traditional techniques such as matched asymptotic expansions, WKB, and multiple scales. Readers unfamiliar with RG methods will find a basic introduction to the method along with three detailed examples: A linear oscillator, a nonlinear Duffing oscillator, and wave propagation in a nonlinear medium. The authors compare the RG solutions to those won through more traditional approaches in popular texts, so that instructors can use this article to augment a standard course on perturbation methods or as supplementary material for eager graduate students.

A fully discrete ε-uniform method for singular perturbation problems on equidistant meshes

We propose a fully discrete ε-uniform finite-difference method on an equidistant mesh for a singularly perturbed two-point boundary-value problem (BVP). We start with a fitted operator method reflecting the singular perturbation nature of the problem through a local BVP. However, to solve the local BVP, we employ an upwind method on a Shishkin mesh in local domain, instead of solving it exactly. Thus, we show that it is possible to develop a ε-uniform method, totally in the context of finite differences, without solving any differential equation exactly. We further study the convergence properties of the numerical method proposed and prove that it nodally converges to the true solution for any ε. Finally, a set of numerical experiments is carried out to validate the theoretical results computationally.

Singularly perturbed advection–diffusion–reaction problems: Comparison of operator-fitted methods

Abstract We compare classical methods for singular perturbation problems, such as El-Mistikawy and Werle scheme and its modifications, to exponential spline collocation schemes. We discuss subtle differences that exist in applying this method to one dimensional reaction–diffusion problems and advection–diffusion problems. For pure advection–diffusion problems, exponential tension spline collocation is less capable of capturing only one boundary layer, which happens when no reaction term is present. Thus the existing collocation scheme in which the approximate solution is a projection to the space piecewisely spanned by {1, x, exp (± px)} is inferior to the generalization of El-Mistikawy and Werle method proposed by Ramos. We show how to remedy this situation by considering projections to spaces locally spanned by {1, x, x 2, exp ( px)}, where p > 0 is a tension parameter. Next, we exploit a unique feature of collocation methods, that is, the existence of special collocation points which yield better global convergence rates and double the convergence order at the knots.

Convergence of parallel multistep hybrid methods for singular perturbation problems

Parallel multistep hybrid methods (PHMs) can be implemented in parallel with two processors, accordingly have almost the same computational speed per integration step as BDF methods of the same order with the same stepsize. But PHMs have better stability properties than BDF methods of the same order for stiff differential equations. In the present paper, we give some results on error analysis of A ( α ) -stable PHMs for the initial value problems of ordinary differential equations in singular perturbation form. Our convergence results are similar to those of linear multistep methods (such as BDF methods), i.e. the convergence orders are equal to their classical convergence orders, and no order reduction occurs. Some numerical examples also confirm our results.

High-Order Compact Finite-Difference Scheme for Singularly-Perturbed Reaction-Diffusion Problems on a New Mesh of Shishkin Type

In this work, a high-order compact finite-difference (HOCFD) scheme has been proposed to solve 1-dimensional (1D) and 2-dimensional (2D) elliptic and parabolic singularly-perturbed reaction-diffusion problems. A new kind of piecewise uniform mesh of Shishkin type (Miller et al. in Fitted Numerical Methods for Singular Perturbation Problems, 1996) has also been proposed and using this mesh the HOCFD scheme gives better results as compared to the results using the Shishkin mesh. Moreover, the stated method gives e-uniform convergence and improved orders of convergence which have also been provided in the results for some test problems.

High order finite volume methods for singular perturbation problems

In this paper we establish a high order finite volume method for the fourth order singular perturbation problems. In conjunction with the optimal meshes, the numerical solutions resulting from the method have optimal convergence order. Numerical experiments are presented to verify our theoretical estimates.

Towards the implementation of the singular function method for singular perturbation problems

The numerical solution of singular perturbation problems (SPPs) is delicate because the perturbation parameter ε and the mesh size h cannot vary independently of one another. In the extended abstract [J.M.-S. Lubuma, K.C. Patidar, Finite element methods for self-adjoint singular perturbation problems, in: T.E. Simos, G. Maroulis (Eds.), ICCMSE 2005: Advances in Computational Methods in Sciences and Engineering, Lecture Series on Computer and Computational Sciences, vol. 4, VSP International Science Publishers, The Netherlands, 2005, pp. 344–347; J.M.-S. Lubuma, K.C. Patidar, Reliable Finite Element Methods for Self-adjoint Singular Perturbation Problems, University of Pretoria Technical Report UPWT 2005/11], the authors proposed the singular function method (SFM) to solve a self-adjoint singular perturbation problem. The SFM is a variant of the finite element method (FEM), where the space of trial and test functions is enriched by the singular functions of the SPP. We proved in the above mentioned work that the SFM is ε-uniformly convergent of optimal order in appropriate norms. However, the numerical implementation of the SFM, based on numerical integration, did not provide satisfactory results. In this paper, we present an alternative way of implementing the SFM. That is, we avoid numerical integration by making use of the explicit form of the singular functions in the computations of the stiffness matrix and the load vector. We obtain improved results. These results are compared with those obtained with numerical integration. The new results confirm theoretical order of convergence.

On Richardson extrapolation for fitted operator finite difference methods

Abstract Recently, there has been a great interest towards the higher order methods for singularly perturbed problems. As compared to their lower order counterparts, they provide better accuracy with fewer mesh points. Construction and/or implementation of direct higher order methods is usually very complicated. Thus a natural choice is to use some convergence acceleration techniques, e.g. Richardson extrapolation. However, as we see in this article, such techniques do not perform equally well on all type of methods. To investigate this, we consider two fitted operator finite difference methods (FOFDMs) developed by Patidar [K.C. Patidar, High order fitted operator numerical method for self-adjoint singular perturbation problems Appl. Math. Comput. 171(1) (2005) 547–566] and Lubuma and Patidar [J. Lubuma, K.C. Patidar, Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems, J. Comput. Appl. Math. 191 (2006) 228–238], referred to as FOFDM-I and FOFDM-II, respectively. The FOFDM-I is fourth and second order accurate for moderate and smaller values of e, respectively. Unfortunately, Richardson extrapolation does not improve the order of this method. The FOFDM-II is second order uniformly convergent and we show that its order can be improved up to four by using Richardson extrapolation. Both the methods are analyzed for convergence and comparative numerical results supporting theoretical estimates are provided.

Fitted fourth-order tridiagonal finite difference method for singular perturbation problems

In this paper, a fitted fourth-order tridiagonal finite difference scheme is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) point. We have taken a fourth-order tridiagonal finite difference scheme by M.M. Chawla [A fourth-order tridiagonal finite difference method for general nonlinear two-point boundary value problems with mixed boundary conditions, J. Inst. Maths Appl. 21 (1978) 83–93] and introduced a fitting factor. The fitting factor is obtained from the theory of singular perturbations. Thomas Algorithm is used to solve the system. To demonstrate the applicability of the present method, we have solved five linear problems (three with left end and two with right end boundary layers). Solutions of these problems using the present fitted method are compared with Chawla’s solutions. From the results, it is observed that the present method is stable and has better approximation to the exact solution.

A seventh order numerical method for singular perturbation problems

In this paper, a seventh order numerical method is presented for solving singularly perturbed two-point boundary value problems with a boundary layer at one end point. The two-point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a seventh order compact difference scheme is presented for the first order system. An asymptotically equivalent first order equation of the original singularly perturbed two-point boundary value problem is obtained from the theory of singular perturbations. It is used in the seventh order compact difference scheme to get a two term recurrence relation and is solved. Several linear and nonlinear singular perturbation problems have been solved and the numerical results are presented to support the theory.