High order parameter uniform numerical method for singular perturbation problems

Patidar, Kailash C.

doi:10.1016/j.amc.2006.10.040

Cited by 21 publications

(14 citation statements)

References 23 publications

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“…Proof See Munyakazi and Patidar (2010a) or Patidar (2007) Using Lemmas 2.3 and 4.1, we obtain Then by Lemma 3.2 and re-instating the dropped time level index, we have proved the following theorem.…”

Section: Lemma 41 For a Fixed Mesh And For All Integers M We Havementioning

confidence: 94%

“…Both these types of methods have been used to solve stationary singularly perturbed problems in one and several dimensions. As examples, see Linß and Stynes (1999), Lubuma andPatidar (2006), Miller et al (1996), Munyakazi and Patidar (2010a,b, 2012), Patidar (2005, 2007, Roos et al (1996), Shishkin (1986. It should be noted that the discovery/development of fitted mesh methods is anterior to that of the fitted operator ones.…”

Section: Introductionmentioning

confidence: 99%

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A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems

Munyakazi

Patidar

2013

Comp. Appl. Math.

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This paper treats a time-dependent singularly perturbed reaction-diffusion problem. We semidiscretize the problem in time by means of the classical backward Euler method. We develop a fitted operator finite difference method (FOFDM) to solve the resulting set of linear problems (one at each time level). We prove that the underlying fitted operator satisfies the maximum principle. This result is then used in the error analysis of the FOFDM. The method is shown to be first order convergent in time and second order convergent in space, uniformly with respect to the perturbation parameter. We test the method on several numerical examples to confirm our theoretical findings. IntroductionWhen solving numerically time-dependent singularly perturbed problems, it is customary to consider dimension splitting: The problems are semidiscretized in time (for example by using the classical backward Euler or Crank-Nicolson scheme). Then, at each time level, a set of stationary problems is solved using a suitably designed numerical method. Singularly perturbed problems involve a (perturbation) parameter which multiplies the highest derivative term in the model-equation of the problem. The solution to such problems is characterized by layer regions which are narrow parts of the domain over which the solution undergoes abrupt changes. It is well known that classical methods are not appropriate when the perturbation parameter becomes small unless very fine meshes are used for spatial discretization. However, this approach has two side effects: it increases the round-off error and the computational cost. There is a vast literature about non-classical numerical methods. In the context of finite differences, we can group these methods into two classes: the class of fitted mesh methods and the class of fitted operator methods. Both these types of methods have been used to solve stationary singularly perturbed problems in one and several dimensions. As examples, see Linß and Stynes (1999), Lubuma andPatidar (2006), Miller et al. (1996), Munyakazi and Patidar (2010a,b, 2012), Patidar (2005, 2007, Roos et al. (1996), Shishkin (1986. It should be noted that the discovery/development of fitted mesh methods is anterior to that of the fitted operator ones. The analysis of the latter is simpler due to the fact that they are based on uniform meshes unlike the former where non-uniform meshes are designed.

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Section: Lemma 41 For a Fixed Mesh And For All Integers M We Havementioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems

Munyakazi

Patidar

2013

Comp. Appl. Math.

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“…On the other hand in Patidar [14] a fitted mesh finite difference method (FMFDM) was shown to be fourth order ε-uniformly convergent. The third order of convergence was found for quasilinear problems in Vulanović [20,21].…”

Section: Introductionmentioning

confidence: 98%

“…To this end, we consider the following problem (1) for which Patidar [14] constructed a fourth order ε-uniformly convergent FMFDM (on a mesh of Shishkin-type):…”

Section: Introductionmentioning

confidence: 99%

Limitations of Richardson’s extrapolation for a high order fitted mesh method for self-adjoint singularly perturbed problems

Munyakazi

Patidar

2009

J. Appl. Math. Comput.

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The aim of this paper is to investigate whether we can accelerate the order of convergence of existing high order methods to solve some singularly perturbed two-point BVPs. To this end, we consider a fitted mesh finite difference method of Patidar (Appl. Math. Comput., 188:720-733, 2007) applied on a mesh of Shishkin type for the solution of self-adjoint problem which is ε-uniformly convergent of order four. We attempted to increase the order of convergence by Richardson's extrapolation and discovered that this well-known convergence acceleration technique has some limitations. We observe that even though this extrapolation technique improves the accuracy slightly, it does not increase the rate of convergence which is originally four for the underlying method for the problem above. Theoretical investigations are demonstrated by some numerical experiments.

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“…[10], Roos et al [19] and the references therein. In the articles [2,[5][6][7]13,[15][16][17]20], many researcher have followed different numerical approach combining fitted mesh methods and fitted operator methods for solving singular perturbation problems where as [8] gives an erudite outline on the numerical methods for singular perturbation problems. In [11,14] efficient numerical methods are used for singularly perturbed differential equations with an delay (or shift) term.…”

Section: Introductionmentioning

confidence: 99%

Exponentially Fitted Finite Difference Scheme For Singularly Perturbed Two Point Boundary Value Problems

Mohapatra

Reddy²

2014

Int. J. Appl. Comput. Math

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In this paper an exponentially fitted finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at left (or right) end of the domain. A fitting factor is introduced and the model equation is discretized by a finite difference scheme on an uniform mesh. Thomas algorithm is used to solve the tri-diagonal system. The stability of the algorithm is investigated. It is shown that proposed technique provides first order accuracy independent of the perturbation parameter. Several linear and nonlinear problems are solved by the proposed method and numerical results are presented to illustrate the theoretical parameter-uniform error bounds established.

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High order parameter uniform numerical method for singular perturbation problems

Cited by 21 publications

References 23 publications

A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems

A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems

Limitations of Richardson’s extrapolation for a high order fitted mesh method for self-adjoint singularly perturbed problems

Exponentially Fitted Finite Difference Scheme For Singularly Perturbed Two Point Boundary Value Problems

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