Application of Sinc-Galerkin method to singularly ... - Semantic Scholar
Numer Algor DOI 10.1007/s11075-013-9752-5 ORIGINAL PAPER
Application of Sinc-Galerkin method to singularly perturbed parabolic convection-diffusion problems J. Rashidinia · A. Barati · M. Nabati
Received: 28 January 2013 / Accepted: 30 July 2013 © Springer Science+Business Media New York 2013
Abstract We develop a numerical algorithm for solving singularly perturbed onedimensional parabolic convection-diffusion problems. The method comprises a standard finite difference to discretize in temporal direction and Sinc-Galerkin method in spatial direction. The convergence analysis and stability of proposed method are discussed in details, it is justifying that the approximate solution converges to the exact solution at an exponential rate. we know that the conventional methods for these problems suffer due to decreasing of perturbation parameter, but the Sinc method handel such difficulty as singularity. This scheme applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behavior of the rates of convergence. Keywords Singularly perturbed convection-diffusion equation · Sinc-Galerkin method · Convergence analysis · Stability Mathematics Subject Classifications (2010) 65M12 · 65M60 · 65M99
J. Rashidinia () · A. Barati · M. Nabati School of Mathematics, Iran University of Science and Technology, Hengam, Narmak, Tehran, Iran e-mail: [email protected] A. Barati e-mail: a [email protected] M. Nabati e-mail: m [email protected]
Numer Algor
1 Introduction We consider the one dimensional parabolic convection-diffusion problem ∂ 2u ∂u ∂u − 2 + a(x, t) + b(x, t)u = f (x, t), ∂t ∂x ∂x (x, t) ∈ ≡ (0, 1) × (0, T ),
(1)
subjected to the initial and boundary conditions : u(x, 0) = u0 (x), x ∈ , u(0, t) = g0 (t), u(1, t) = g1 (t),
(2)
where is a diffusion coefficient or singular perturbation parameter satisfying 0 < 1, a(x, t) is the velocity and f (x, t) − b(x, t)u is the reaction term. We assume that the functions a(x, t), b(x, t), f (x, t) and u0 (x) are sufficiently smooth, the solution of problem (1) will be smooth on whole of domain , and moreover condition b(x, t) ≥ 0 ensures the uniqueness of the solution. Also, the boundary layer is located at x = 1 if a(x, t) > 0 and b(x, t) ≥ 0. (Roos et al. [19]) Convection-diffusion equations are important in many branches of engineering, physics and applied sciences. These equations describe the transport of solute in groundwater and surface water, the displacement of oil by fluid injection in oil recovery, the movement of aerosols and trace gases in the atmosphere, and miscible fluid flow processes in many other applications. The peculiarity of this equation is that it represents the coupling of two different phenomena, convection and diffusion. It also serves as a simplified model problem to the Navier-Stokes equation in fluid dynamics. Numerical treatment of problem (1) has been wide spread in recent years. As becomes small, the solutions to such problem display, near the boundary, thin transition layers called boundary layers Roos et al. [19]. The presence of the singular perturbation parameter , leads to occurrences of spurious oscillations in the computed solutions using classical central finite difference schemes and finite element methods with piecewise polynomial basis functions. Therefore, in order to overcome such drawbacks associated with classical finite difference and finite element methods, we need to develop -uniformly convergent numerical methods; among them the fitted mesh method, which utilizes special layer-adapted mesh, is a satisfactory and popular technique to overcome the numerical difficulties. In recent years, for singularly perturbed parabolic convection-diffusion problems, a variety of uniformly convergent numerical methods based on Shishkin meshes have been developed. For instant, Kadalbajoo et al. [9] provided a uniform convergent numerical method with respect to the diffusion parameter, they applied the implicit Euler method for the time discretization and B-spline collocation method on nonuniform mesh for the spatial direction. Clavero et al. [3] gave a uniform convergent with respect to based on implicit Euler method for the time discretization and the simple upwind finite difference scheme on the Shishkin mesh for the spatial discretization. Also Clavero et al. [5] used the second-order HODIE scheme with uniform convergent in for time-dependent convection-diffusion problems. Mukherjee and Natesan [13] derived a uniformly convergent hybrid numerical scheme attaining order of convergence one in time and almost two in space. Ramos [18] presented an exponentially fitted method and showed its uniform convergence in the perturbation
Numer Algor
parameter. Moreover, we can point to many other efficient methods for solving singularly perturbed parabolic convection-diffusion problems such as Kadalbajoo et al. [10], O’Riordan et al. [16], Natesan and Deb [7], Cai and Liu [2] and M. EL-Gamel [8]. In this paper, we apply the Sinc-Galerkin method to solve singularly perturbed parabolic convection-diffusion problems. At the first stage, our method is based on the discretization of the time variable by means of the implicit Euler method and freezing the coefficients of the resulting ordinary differential equation at each time step. At the second stage, we use the Sinc-Galerkin method on the yield linear ordinary differential equations at each time step resulting from the time semidiscretization. In the Sinc method the test functions are defined by the Sinc-function s(x) = sin(πx)/(πx), the Sinc method, which was developed by Stenger [20], is based on the Whittaker-Shannon-Kotel’ nikov sampling theorem for entire functions. This method has many advantages over classical methods that use polynomials as bases. For example, in the presence of singularities, it gives a much better rate of convergence and accuracy than polynomial methods. In recent years, a lot of attention has been devoted to the study of the Sinc-Galerkin method to investigate various scientific models. The efficiency of the Sinc method has been formally proved by many researchers Bialecki [1], Rashidinia and Zarebnia [17, 22], Nurmuhammada et al. [14], EL-Gamel [8], Saadatmandi, Dehghan [21] and Okayama et al. [15]. The paper is organized as follows. In Section 2, we review some basic facts about the sinc approximation. In Section 3, we discretized the temporal variable by means of implicit Euler method. In Section 4, the Sinc-Galerkin method is developed for solving of the arising second-order singularly perturbed boundary value problems with homogeneous boundary condition. In Section 5, the convergence analysis of proposed method is given, in this section our contribution is mainly due to proof of the Lemma 2 and Theorem 3. Also in Section 6, the stability of method is discussed. Some numerical examples will be presented in Section 7, and at the end we conclude implementation, application and efficiency of proposed scheme.
2 Notation and background In this section, we state preliminaries of the Sinc interpolation together with some essential definitions and theorems. The Sinc function is defined on −∞ < x < ∞ by Sinc(x) =
sin(πx) πx ,
1,
x = 0, x = 0.
For h > 0 we will denote the Sinc basis functions by S(j, h)(x) = sinc
x − jh , h
j = 0, ±1, ±2, . . .
Numer Algor
let f be a function defined on R then for h > 0 the series C(f, h)(x) =
∞
f (j h)S(j, h)(x),
j =−∞
is called the Whittaker cardinal expansion of f whenever this series converges. The properties of Whittaker cardinal expansions have been studied and are thoroughly surveyed in Stenger [20]. These properties are derived in the infinite strip Dd of the complex plane where d > 0 π Dd = ζ = ξ + iη : |η| < d ≤ . 2 Approximations can be constructed for infinite, semi-finite, and finite intervals. But in this paper we construct approximation on the interval (0, 1),we consider the conformal map z φ(z) = ln , (3) z−1 which maps the eye-shaped region z π DE = z = x + iy; | arg |
Application of Sinc-Galerkin method to singularly perturbed parabolic convection-diffusion problems J. Rashidinia · A. Barati · M. Nabati
Received: 28 January 2013 / Accepted: 30 July 2013 © Springer Science+Business Media New York 2013
Abstract We develop a numerical algorithm for solving singularly perturbed onedimensional parabolic convection-diffusion problems. The method comprises a standard finite difference to discretize in temporal direction and Sinc-Galerkin method in spatial direction. The convergence analysis and stability of proposed method are discussed in details, it is justifying that the approximate solution converges to the exact solution at an exponential rate. we know that the conventional methods for these problems suffer due to decreasing of perturbation parameter, but the Sinc method handel such difficulty as singularity. This scheme applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behavior of the rates of convergence. Keywords Singularly perturbed convection-diffusion equation · Sinc-Galerkin method · Convergence analysis · Stability Mathematics Subject Classifications (2010) 65M12 · 65M60 · 65M99
J. Rashidinia () · A. Barati · M. Nabati School of Mathematics, Iran University of Science and Technology, Hengam, Narmak, Tehran, Iran e-mail: [email protected] A. Barati e-mail: a [email protected] M. Nabati e-mail: m [email protected]
Numer Algor
1 Introduction We consider the one dimensional parabolic convection-diffusion problem ∂ 2u ∂u ∂u − 2 + a(x, t) + b(x, t)u = f (x, t), ∂t ∂x ∂x (x, t) ∈ ≡ (0, 1) × (0, T ),
(1)
subjected to the initial and boundary conditions : u(x, 0) = u0 (x), x ∈ , u(0, t) = g0 (t), u(1, t) = g1 (t),
(2)
where is a diffusion coefficient or singular perturbation parameter satisfying 0 < 1, a(x, t) is the velocity and f (x, t) − b(x, t)u is the reaction term. We assume that the functions a(x, t), b(x, t), f (x, t) and u0 (x) are sufficiently smooth, the solution of problem (1) will be smooth on whole of domain , and moreover condition b(x, t) ≥ 0 ensures the uniqueness of the solution. Also, the boundary layer is located at x = 1 if a(x, t) > 0 and b(x, t) ≥ 0. (Roos et al. [19]) Convection-diffusion equations are important in many branches of engineering, physics and applied sciences. These equations describe the transport of solute in groundwater and surface water, the displacement of oil by fluid injection in oil recovery, the movement of aerosols and trace gases in the atmosphere, and miscible fluid flow processes in many other applications. The peculiarity of this equation is that it represents the coupling of two different phenomena, convection and diffusion. It also serves as a simplified model problem to the Navier-Stokes equation in fluid dynamics. Numerical treatment of problem (1) has been wide spread in recent years. As becomes small, the solutions to such problem display, near the boundary, thin transition layers called boundary layers Roos et al. [19]. The presence of the singular perturbation parameter , leads to occurrences of spurious oscillations in the computed solutions using classical central finite difference schemes and finite element methods with piecewise polynomial basis functions. Therefore, in order to overcome such drawbacks associated with classical finite difference and finite element methods, we need to develop -uniformly convergent numerical methods; among them the fitted mesh method, which utilizes special layer-adapted mesh, is a satisfactory and popular technique to overcome the numerical difficulties. In recent years, for singularly perturbed parabolic convection-diffusion problems, a variety of uniformly convergent numerical methods based on Shishkin meshes have been developed. For instant, Kadalbajoo et al. [9] provided a uniform convergent numerical method with respect to the diffusion parameter, they applied the implicit Euler method for the time discretization and B-spline collocation method on nonuniform mesh for the spatial direction. Clavero et al. [3] gave a uniform convergent with respect to based on implicit Euler method for the time discretization and the simple upwind finite difference scheme on the Shishkin mesh for the spatial discretization. Also Clavero et al. [5] used the second-order HODIE scheme with uniform convergent in for time-dependent convection-diffusion problems. Mukherjee and Natesan [13] derived a uniformly convergent hybrid numerical scheme attaining order of convergence one in time and almost two in space. Ramos [18] presented an exponentially fitted method and showed its uniform convergence in the perturbation
Numer Algor
parameter. Moreover, we can point to many other efficient methods for solving singularly perturbed parabolic convection-diffusion problems such as Kadalbajoo et al. [10], O’Riordan et al. [16], Natesan and Deb [7], Cai and Liu [2] and M. EL-Gamel [8]. In this paper, we apply the Sinc-Galerkin method to solve singularly perturbed parabolic convection-diffusion problems. At the first stage, our method is based on the discretization of the time variable by means of the implicit Euler method and freezing the coefficients of the resulting ordinary differential equation at each time step. At the second stage, we use the Sinc-Galerkin method on the yield linear ordinary differential equations at each time step resulting from the time semidiscretization. In the Sinc method the test functions are defined by the Sinc-function s(x) = sin(πx)/(πx), the Sinc method, which was developed by Stenger [20], is based on the Whittaker-Shannon-Kotel’ nikov sampling theorem for entire functions. This method has many advantages over classical methods that use polynomials as bases. For example, in the presence of singularities, it gives a much better rate of convergence and accuracy than polynomial methods. In recent years, a lot of attention has been devoted to the study of the Sinc-Galerkin method to investigate various scientific models. The efficiency of the Sinc method has been formally proved by many researchers Bialecki [1], Rashidinia and Zarebnia [17, 22], Nurmuhammada et al. [14], EL-Gamel [8], Saadatmandi, Dehghan [21] and Okayama et al. [15]. The paper is organized as follows. In Section 2, we review some basic facts about the sinc approximation. In Section 3, we discretized the temporal variable by means of implicit Euler method. In Section 4, the Sinc-Galerkin method is developed for solving of the arising second-order singularly perturbed boundary value problems with homogeneous boundary condition. In Section 5, the convergence analysis of proposed method is given, in this section our contribution is mainly due to proof of the Lemma 2 and Theorem 3. Also in Section 6, the stability of method is discussed. Some numerical examples will be presented in Section 7, and at the end we conclude implementation, application and efficiency of proposed scheme.
2 Notation and background In this section, we state preliminaries of the Sinc interpolation together with some essential definitions and theorems. The Sinc function is defined on −∞ < x < ∞ by Sinc(x) =
sin(πx) πx ,
1,
x = 0, x = 0.
For h > 0 we will denote the Sinc basis functions by S(j, h)(x) = sinc
x − jh , h
j = 0, ±1, ±2, . . .
Numer Algor
let f be a function defined on R then for h > 0 the series C(f, h)(x) =
∞
f (j h)S(j, h)(x),
j =−∞
is called the Whittaker cardinal expansion of f whenever this series converges. The properties of Whittaker cardinal expansions have been studied and are thoroughly surveyed in Stenger [20]. These properties are derived in the infinite strip Dd of the complex plane where d > 0 π Dd = ζ = ξ + iη : |η| < d ≤ . 2 Approximations can be constructed for infinite, semi-finite, and finite intervals. But in this paper we construct approximation on the interval (0, 1),we consider the conformal map z φ(z) = ln , (3) z−1 which maps the eye-shaped region z π DE = z = x + iy; | arg |
