Accurate and approximate analytic solutions of singularly perturbed differential equations with two-dimensional boundary layers

Li, Zi‐Cai; Tsai, Heng-Shuing; Wang, Song; Miller, John J. H.

doi:10.1016/j.camwa.2007.10.027

Cited by 4 publications

(4 citation statements)

References 7 publications

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“…Then the true solution of (5.1)–(5.4) is found in by

\begin{matrix} left \\ u (x, y) = \frac{\sinh (\frac{α}{2 ε} x)}{\sinh (\frac{α}{2 ε} π)} \exp (- \frac{α (π - x)}{2 ε}) + \frac{\sinh (\frac{β}{2 ε} y)}{\sinh (\frac{β}{2 ε} π)} \exp (- \frac{β (π - y)}{2 ε}) \\ - \frac{\sinh (\frac{α}{2 ε} x) \sinh (\frac{β}{2 ε} y)}{\sinh (\frac{α}{2 ε} π) \sinh (\frac{β}{2 ε} π)} \exp (\frac{- α (π - x) - β (π - y)}{2 ε}) . \end{matrix}

…”

Section: Numerical Experiments and Concluding Remarksmentioning

confidence: 99%

“…Choose Model C in Li et al ,

£ u \equiv - ε (u_{x x} + u_{y y}) + α u_{x} + β u_{y} = 0 in S, u = g on Γ,

where

α \geq β > 0

, and

S = {(x, y) | 0 < x < π, 0 < y < π}

. The Dirichlet condition on Γ is designed deliberately as

u (x, π) = u (π, y) = 1,

u (x, 0) = \frac{\sinh (\frac{α}{2 ε} x)}{\sinh (\frac{α}{2 ε} π)} \exp (- \frac{α (π - x)}{2 ε}),

u (0, y) = \frac{\sinh (\frac{β}{2 ε} y)}{\sinh (\frac{β}{2 ε} π)} \exp (- \frac{β (π - y)}{2 ε}) .

…”

Section: Numerical Experiments and Concluding Remarksmentioning

confidence: 99%

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Stability analysis for singularly perturbed differential equations by the upwind difference scheme

Wei

Huang

et al. 2013

Numerical Methods Partial

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For solving singularly perturbed differential equations (SPDE), the upwind difference scheme (UDS) and the fitted difference method are chosen. The local refinements of grids are adopted in singular layers with the minimal meshspacing h min = O(hε), where h is the maximal meshspacing, and ε( 1) a very small parameter. The traditional condition number in 2-norm is given by Cond = O(h −2 ε −1 ). For the infinitesimal ε(≤ 10 −8 ) in application, the huge Cond issues a dilemma regarding whether the numerical solutions by the UDS can be trusted. Although the UDS has been used for several decades, such a dilemma has not been clarified yet. The goal of this article is to clarify this dilemma. To this end, we solicit the effective condition number Cond_eff in Li et al. Numer Linear Algebra Appl 15 (2008), 575-690 Effective condition for Numerical Partial Different Equation, 2013, and develop a new actual condition number from the maximum principle. Both of them may offer much smaller bounds of the solution errors caused by perturbation, e.g., rounding, truncation, or discritization errors. We study the Dirichlet problems of SPDE by the UDS in a rectangle. When the Dirichlet boundary condition on the downwind side is homogeneous, we derive Cond_eff = O( 1 √ h ). When the entire Dirichlet boundary conditions are homogeneous, the extraordinary bound, Cond_eff = O(1), is achieved. Moreover, we derive the actual condition numbers as Cond_actual = O( 1 √ h ) and Cond_actual = O( 1 h ) for the homogeneous and the nonhomogeneous SPDE, respectively. Note that these bounds do not depend on ε; this is distinct from the traditional Cond. Based on the analysis of this article, the existing dilemma caused by Cond has been removed, to grant a good stability of the UDS for SPDE.

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“…Then the true solution of (5.1)–(5.4) is found in by

\begin{matrix} left \\ u (x, y) = \frac{\sinh (\frac{α}{2 ε} x)}{\sinh (\frac{α}{2 ε} π)} \exp (- \frac{α (π - x)}{2 ε}) + \frac{\sinh (\frac{β}{2 ε} y)}{\sinh (\frac{β}{2 ε} π)} \exp (- \frac{β (π - y)}{2 ε}) \\ - \frac{\sinh (\frac{α}{2 ε} x) \sinh (\frac{β}{2 ε} y)}{\sinh (\frac{α}{2 ε} π) \sinh (\frac{β}{2 ε} π)} \exp (\frac{- α (π - x) - β (π - y)}{2 ε}) . \end{matrix}

…”

Section: Numerical Experiments and Concluding Remarksmentioning

confidence: 99%

“…Choose Model C in Li et al ,

£ u \equiv - ε (u_{x x} + u_{y y}) + α u_{x} + β u_{y} = 0 in S, u = g on Γ,

where

α \geq β > 0

, and

S = {(x, y) | 0 < x < π, 0 < y < π}

. The Dirichlet condition on Γ is designed deliberately as

u (x, π) = u (π, y) = 1,

u (x, 0) = \frac{\sinh (\frac{α}{2 ε} x)}{\sinh (\frac{α}{2 ε} π)} \exp (- \frac{α (π - x)}{2 ε}),

u (0, y) = \frac{\sinh (\frac{β}{2 ε} y)}{\sinh (\frac{β}{2 ε} π)} \exp (- \frac{β (π - y)}{2 ε}) .

…”

Section: Numerical Experiments and Concluding Remarksmentioning

confidence: 99%

Stability analysis for singularly perturbed differential equations by the upwind difference scheme

Wei

Huang

et al. 2013

Numerical Methods Partial

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show abstract

“…Our finite point method has been tailored based on the local exponential basis functions. [3,14,15,16,20,22]). We also study the error estimates of our TFPM.…”

mentioning

confidence: 99%

“…Recently, the typical methods for high dimensional problems are proposed to solve such singular perturbation problems based on the refined mesh technique, such as Shishkin meshes [3,14,15,16,20,22], in which people can get the uniform convergence on nonuniform meshes. But for high dimensional problems, people did not get the uniform convergence results by such technique.…”

mentioning

confidence: 99%

Tailored finite point method based on exponential bases for convection-diffusion-reaction equation

Han

Huang

2012

Math. Comp.

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In this paper, we propose a class of new tailored finite point methods (TFPM) for the numerical solution of a type of convection-diffusionreaction problems in two dimensions. Our finite point method has been tailored based on the local exponential basis functions. Furthermore, our TFPM satisfies the discrete maximum principle automatically. We also study the error estimates of our TFPM. We prove that our TFPM can achieve good accuracy even when the mesh size h ε for some cases without any prior knowledge of the boundary layers. Our numerical examples show the efficiency and reliability of our method.Kellogg [2] studied some one-dimensional problems, they used the method of "exponential fitting" to get the uniformly convergent approximation. But for high dimensional problems, people did not get the uniform convergence results by such technique. Recently, the typical methods for high dimensional problems are proposed to solve such singular perturbation problems based on the refined mesh technique, such as Shishkin meshes [3,14,15,16,20,22], in which people can get the uniform convergence on nonuniform meshes. Melenk et al. also used so-called hp-version finite element method to achieve uniform convergence (cf.[17] and the references in it). For more related work, please refer to the books by Morton [19], Roos, Stynes and Tobiska [21], and the review paper by Stynes [23]. Generally speaking, in those methods for high dimensional problems, one usually needs the finest mesh size h ∼ O(ε) to achieve a satisfied numerical result. Therefore, most of those methods need many degrees of freedom when ε 1, although they are uniformly convergent, especially if they want to have small pointwise errors on all mesh points. For example, by Shishkin meshes, one can get the uniform convergence rate, but they need a prior knowledge of the position of boundary layer (cf. [3,14,15,16,20,22]).Recently, we develop an efficient method, so-called "tailored finite point method ", for the numerical simulation of singular perturbation problems and high frequency wave propagation problems [6,7,8,9,11]. In our work, we proposed a kind of finite point methods which used the exact solutions of the local approximation problems to construct the global approximate solution. Therefore, we can have high accuracy even on the uniform coarse mesh h ε without any prior knowledge of the boundary layers. In [7], we have proposed some finite point schemes for convection-diffusion problems using Bessel functions. In that paper, our schemes also obtained very good approximate solutions on coarse mesh, but those schemes did not satisfy the discrete maximum principle in some cases.In this paper, we propose a class of new tailored finite point methods to solve the problem (1.1)-(1.2) numerically, in which we use the proper exponential functions to construct the approximate solution. These exponential functions are the exact solution of the proper local approximate problems. We prove that these new schemes can always satisfy the discrete maximum principle, we also...

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A brief survey on numerical methods for solving singularly perturbed problems

Kadalbajoo¹,

Gupta²

2010

Applied Mathematics and Computation

126

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Accurate and approximate analytic solutions of singularly perturbed differential equations with two-dimensional boundary layers

Cited by 4 publications

References 7 publications

Stability analysis for singularly perturbed differential equations by the upwind difference scheme

Stability analysis for singularly perturbed differential equations by the upwind difference scheme

Tailored finite point method based on exponential bases for convection-diffusion-reaction equation

A brief survey on numerical methods for solving singularly perturbed problems

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