Finite volume element method for analysis of unsteady reaction diffusion problems

Transcription

1 Acta Mech Sin (2009) 25: DOI /s RESEARCH PAPER Finite volume element method for analysis of unsteady reaction diffusion problems Sutthisak Phongthanapanich Pramote Dechaumphai Received: 6 May 2008 / Revised: 6 October 2008 / Accepted: 5 December 2008 / Published online: 27 February 2009 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2009 Abstract A finite volume element method is developed for analyzing unsteady scalar reaction diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the highgradient boundary layers. Keywords Finite volume element method Explicit method Unsteady problem Singularly perturbed equation Reaction diffusion 1 Introduction Recent development of several reaction diffusion analysis schemes in biology, ecology, biochemistry, fluid mechanics S. Phongthanapanich Department of Mechanical Engineering Technology, College of Industrial Technology, King Mongkut s University of Technology North Bangkok, Bangkok 10800, Thailand sutthisakp@kmutnb.ac.th P. Dechaumphai (B) Mechanical Engineering Department, Chulalongkorn University, Bangkok 10330, Thailand fmepdc@eng.chula.ac.th and heat transfer has led to physically interesting and mathematically challenging problems related to the nontrivial numerical solution of the singularly perturbed initial/boundary value problem [1 9]. In general, solution of the singularly perturbed problem contains high-gradient boundary layers along the boundary of the domain [2]. Numerical approximation of such boundary layer flow behavior is not simple, and thus the development of accurate numerical method for analyzing this problem must be done carefully. Some examples of such difficulty are encountered in the cases where the reaction diffusion equation cannot be satisfied by a globally smooth function. In particular, if the unknown scalar variables vary rapidly within a thin layer, many standard numerical schemes may lead to inaccurate and unstable solutions [8 10]. In particular, a robust numerical method, in the sense that the error should not deteriorate as the singular perturbation parameter tends to zero, is required. During the past decade, many techniques of the finite element method for solving the singularly perturbed problems have been proposed and investigated [2,4,7,8,11,12]. The conventional Galerkin finite element method usually fails to analyze such unsteady reaction diffusion problems when either the diffusion coefficient becomes small or a small time step is employed in the time discretization. The stabilized finite element method overcame this trouble by employing the balanced residual perturbation term with some stabilization parameters. However, such method has great influence on the solution accuracy [13]. An improved finite element method so-called the Petrov Galerkin enriched method (PGEM), where the trial space is enriched by the multiscale functions, is later proposed by Franca et al. [12,14]. The method provides good solution accuracy as compared to the other finite element methods for solving the scalar reaction diffusion equation. Some other finite element methods, such as those use the Shishkin mesh, have also been

2 482 S. Phongthanapanich, P. Dechaumphai developed [2 4]. These methods apply robust algebraic error bounds, or employ the p/hp version of the finite element method on appropriate meshes [15,16]. At the same time, some researchers employed the finite difference algorithm to analyze the reaction diffusion problems [5,17]. Liao et al. proposed an efficient higher-order finite difference algorithm for solving the system of twodimensional reaction diffusion equations with nonlinear reaction term. The Padé approximation and Richardson extrapolation are used to achieve high-order solution accuracy in both the spatial and temporal domains. El Salam and Shehata [5] used the implicit Crank Nicolson technique to linearize the governing equations and then solved the system of algebraic equations by using the LU-decomposition method. Recently, many researchers proposed some mixed techniques of the finite element and finite volume methods for analyzing the parabolic equation [18 20]. Rui [18] presented a mixed covolume method for solving the parabolic equation from the general self-adjoint parabolic problem using a variable nondiagonal diffusion tensor and the lowest order Raviart Thomas mixed element. The basic idea of creating a covolume method is to seek for a good combination of the finite volume method and the MAC scheme. Ma et al. [19] proposed the semi-discrete and full-discrete symmetric finite volume schemes based on the linear finite elements for a class of parabolic problems. The symmetric finite volume schemes are analyzed on dual mesh (cell-vertex scheme) of triangular cells. Recently, Wang [20] proposed three schemes of the finite volume element method for solving two-dimensional parabolic equations. His first scheme is analogous to Douglas finite difference scheme with second-order splitting error; the second scheme has third-order splitting error; and the third scheme is an extended locally one-dimensional scheme. The finite volume schemes are also discretized on dual mesh and the nodal value at the center of control volume is approximated by linear finite element interpolation functions. It should be noted that the computational techniques for analyzing the hyperbolic equation are generally classified into the explicit and implicit (or semi-implicit) methods. The explicit method is a popular one because it is simple and requires less computer memory. However, the method is constrained by the CFL condition in order to stabilize the spatial error from growing without bound. Attempts to relief such constraint have been developed, such as one proposed by Wang and Liu [21]. On the other hand, the implicit method provides a more stable solution. However, large time step size may not be used because the solution accuracy degrades with time. The inversion of the coefficient matrix is another weakness of the method because it is a time consumable process. In addition, a large block of memory is required for the coefficient matrix formation. A finite volume element method, that combines the finite volume and the finite element methods, is proposed for analyzing the unsteady scalar reaction diffusion equation in this paper. The cell-centered finite volume method is used to analyze the problem on a uniform control volume. The concept of the finite element method is applied to estimate the gradient quantities herein by employing the method of weighted residuals to determine the nodal gradient quantities. This approach is different from those presented in Refs. [18 20], where the distribution of gradient quantities on element is simply approximated by using linear finite element interpolation functions. Finally, the trapezoidal rule is used to calculate the gradient quantities at cell faces. The robustness and accuracy of the combined method are examined by using reaction diffusion problems. The presentation of the paper starts from explanation of the theoretical formulation in Sect. 2. The proposed method is then evaluated in Sect. 3 by using three examples. The numerical results from these examples are used to evaluate the applicability of the combined method for analyzing both the steady and unsteady reaction diffusion problems. 2 Finite volume element formulation In this paper, the finite volume approach is used to discretize the following two-dimensional scalar reaction diffusion equation. φ ɛ 2 φ + κφ = f, in T, (1) t subject to the boundary conditions φ = g D, on D, (2) ɛ φ n = g N, on N, (3) with = D N with D N =. The initial condition is defined for x with R 2 by φ(x, 0) = φ 0 (x), where φ is the unknown scalar quantity, ɛ>0isthe diffusion coefficient, κ is the reaction coefficient, f = f (x) is the prescribed source term, and t (0, T ) for T <. The computational domain is first discretized into a collection of non-overlapping rectangular control volumes, i = 1,...,N, that completely cover the domain such that = N, =, and j =, if i = j. i=1 Equation (1) is then integrated over the arbitrary control volume to obtain ( ) φ ɛ 2 φ + κφ f dx = 0. (5) t (4)

3 Finite volume element method for analysis of unsteady reaction diffusion problems 483 The finite difference approximation and the divergence theorem are then applied to the temporal and spatial terms, respectively, of the above equation to yield φ(x, t n+1 )dx = φ(x, t n )dx + t ɛ κφ(x, t)dx + n i (ν) φ(ν,t)dν f (x)dx, (6) where n i (ν) is the unit normal vector of. In the conventional finite volume method, the approximation to the cell average of φ over control volume at time t n and t n+1 is represented by φi n+1 = 1 φ(x, t n+1 )dx, (7) φi n = 1 φ(x, t n )dx, where is the measure of. For an arbitrary control volume, the flux integral over appearing on the right-hand side of Eq. (6) could be approximated by the summation of fluxes passing through all the adjacent cell faces. Hence, by applying the quadrature integration formula on the spatial domain term, the flux integral over may be approximated by ɛ n i (ν) φ(ν,t)dν = ɛ 4 Ɣij nij φij n, (8) j=1 where Ɣ ij is the segment of boundary between the two adjacent cells and j, which is defined by = 4j=1 Ɣ ij and Ɣ ij = j. The integrations of the reaction and the source terms could be approximated by the mid-point quadrature rule to obtain κφ(x, t)dx = κφi n, (9) f (x)dx = f (x i ). (10) By substituting Eqs. (7) (10) into Eq. (6), a fully explicit finite volume scheme for solving Eq. (1) can be written in the form φ n+1 i = φ n i + ɛ t 4 Ɣij nij φij n t(κφn i f i ). j=1 (11) For unsteady computation, the second-order temporal accuracy is achieved by implementing the second-order accurate Runge Kutta time stepping method [22]. The treatment of boundary conditions in this paper is based on using the concept of ghost cell. The central differencing at the boundary is used for the Neumann boundary condition, while the Dirichlet boundary condition is prescribed at the center of the ghost cell. In this paper, the gradient term, φij n, is approximated by the weighted residuals method which is commonly used in the finite element technique [23]. Firstly, the φi n is assumed to distribute linearly over cell in the form φ n i = 4 N k (x) φk n, (12) k=1 where N k (x) denotes the linear interpolation functions for rectangular cell. By applying the weighted residuals method and the Gauss s theorem to Eq. (12), the gradient quantities at point J are obtained as φ n J,i = M 1 n i (ν)n J (ν)φi n dν N J (x) x φi n dx, (13) where M is the consistent mass matrix [23], and φ n J,i are the contributions of the gradient quantities in the cell to the gradient quantities at point J. As an example, Fig. 1 shows four rectangular cells surrounding point J. In order to compute the total gradient quantities at point J, Eq.(13) is applied to all cells surrounding the point J such that φ n J = I φ n J,i, (14) i=1 Fig. 1 Rectangular cells surrounding point J

4 484 S. Phongthanapanich, P. Dechaumphai where I is the number of the surrounding cells. Finally, the gradient quantities at Ɣ ij, φij n, are then computed by applying the trapezoidal rule along the common edge of cells i and j. Finally, for simplicity, the order of accuracy and stability of the explicit numerical scheme [24] given by Eq. (11) will be analyzed on uniform one-dimensional grid cell, = x. The numerical equation for ith cell, (x i 1/2, x i+1/2 ), may be written as following φ n+1 i = φ n i + ɛ t x ( φ x n i+1/2 φ x n i 1/2 t ( κφ n i f i ). (15) By using one-dimensional linear interpolation function, the gradient quantities at cell faces i 1/2 and i + 1/2 can be written by φ x φ x n i 1/2 n i+1/2 = 1 x (φn i φ n i 1 ), = 1 x (φn i+1 φn i ), ) (16) and by using these expressions on the right hand side of Eq. (15) togive φ n+1 i = φ n i + α(φ n i+1 2φn i + φ n i 1 ) t ( κφ n i f i ), (17) where α = ɛ t/( x) 2. Then, truncation error analysis by applying the Taylors series expansion on Eq. (17) evaluated at (x n, t n ) = (i, n) to provide that the order of accuracy of Eq. (17) with the second-order accurate Runge Kutta time stepping method is O( t 2, x 2 ). To analyze the stability of the numerical scheme, the discrete Fourier transform is applied to the homogeneous part of Eq. (17), term by term, to obtain φ n+1 =[1 2α(1 cos(ξ)) κ t] φ n [ ( ) ] ξ = 1 4α sin 2 κ t φ n. (18) 2 By choosing α<1/2, the amplification factor is given by ( ) ξ 1 4α sin2 κ t 2 eκ t. (19) Fig. 2 Numerical solutions on grid size at four different times of problem 3.1: a t = 0.25, b t = 0.50, c t = 0.75, d t = 1.0

5 Finite volume element method for analysis of unsteady reaction diffusion problems 485 Lastly, by applying the result of Eq. (18) n + 1 times and using the Parseval s Identity, to yield φ n+1 2, x e κ(n+1) t φ 0 2, x. (20) Hence by the definition of stability [24], Eq. (17) is stable with respect to the discrete L 2 norm. 3 Numerical examples To evaluate the robustness and accuracy of the proposed finite volume element method, three reaction diffusion problems subjected to homogeneous boundary conditions are examined. The first example is used to evaluate the capability of the proposed method for the singularly perturbed problem where the diffusion coefficient is very small as compared to the reaction coefficient. In the second example, the diffusion coefficient is given as one in order to examine the robustness of the method when a small time step is employed. The numerical results obtained from these two examples will be compared with the finite element solutions presented by Franca et al. [12]. The last example is similar to the first one but with non-zero homogeneous boundary conditions. The accuracy of the numerical solutions obtained from this example will be compared with the finite element solutions presented by Franca et al. [13]. 3.1 Reaction-dominated diffusion problem The first example is a reaction-dominated diffusion problem, with ɛ = 10 6, κ = 1, and f = 1. The initial condition is given by φ 0 (x) = 0. A unit square domain is divided into 400 uniform square grids (20 20). The problem is analyzed using the fixed time step of t = 0.1. Figure 2a d show the numerical solutions of the tested grids at four different times. The four computed profiles show that there is no spurious oscillation along the high-gradient boundary layers. For the purpose of comparison with the solutions presented in Ref. [12] (variant of the PGEM), the analysis is repeated but at different times of t = 0.1, and t = 7.5. At time t = 0.1, the computed profiles and the solutions at x = 0.5 are shown in Fig. 3a and b, respectively. Figure 4a and b show the similar behavior of the solution at the time of t = 7.5. Spurious oscillation does not occur along the high-gradient boundary layers of the solution profiles at these two times. The maximum values are comparable with the solutions from the method presented by Ref. [12]. Fig. 3 Numerical solution of problem 3.1: a At time t = 0.1, b along the line x = 0.5 Fig. 4 Numerical solution of problem 3.1: a At time t = 7.5, b along the line x = 0.5

6 486 S. Phongthanapanich, P. Dechaumphai Fig. 5 Numerical solutions on grid size at four different times of problem 3.2: a t = 0.01, b t = 0.05, c t = 0.1, d t = 1.0 Fig. 6 Numerical solution of problem 3.2: a At time t = 10 5, b along the line x = Reaction diffusion problem The second example is a reaction diffusion problem, with ɛ = 1, κ = 1, and f = 1. The initial condition is given by φ 0 (x) = 0, and the unit square domain is divided into 400 uniform square grids (20 20) in the same fashion as shown in the first example. The objective of this second example is to examine the robustness of the method using a small time step value of t = Figure 5a d shows the computed solutions of the tested grids at four different times of t = 0.01, 0.05, 0.1, and 1.0, respectively. The analysis of this problem is performed again at the two different times of t = 10 5, and t = 10 3, so that the solutions can be compared with the finite element solutions from Ref. [12]. At the time t = 10 5, the computed profiles and the solutions at x = 0.5 are shown in Fig. 6a and b, respectively. There is no spurious oscillation of the computed solution profiles along the high-gradient boundary layers at this earlier time. Figure 7a and b shows the same solution behavior but at the time t = 10 3.Again, the computed maximum values are comparable with the solutions from the method proposed by Ref. [12]. The present method provided less diffusive than the Variant of the PGEM.

7 Finite volume element method for analysis of unsteady reaction diffusion problems 487 Fig. 7 Numerical solution of problem 3.2: a At time t = 10 3, b along the line x = 0.5 Fig. 8 Numerical solutions on grid size at four different times of problem 3.3: a t = 0.25, b t = 0.50, c t = 0.75, d t = 1.0

8 488 S. Phongthanapanich, P. Dechaumphai Fig. 9 Numerical solution of problem 3.3: a At time t = 7.5, b along the line x = Reaction-dominated diffusion problem with non-zero boundary conditions The parameters for the last example are given by ɛ = 10 6, κ = 1, and f = 1. The initial condition is given by φ 0 (x) = 0. The boundary conditions along x = 0 and y = 1 are set as one, while the remaining boundaries are specified as zero. The computational domain and time step used in the first example are also used herein. Figure 8a d shows the numerical solutions of the tested grid at four different times of t = 0.25, 0.50, 0.75, and 1.0, respectively. The computed profiles show that there is no spurious oscillation along highgradient boundary layers. Again, for the purpose of comparison, the steady-state solution profile at time t = 7.5 is shown in Fig. 9a, and the corresponding solution at x = 0.5 are presented in Fig. 9b. These figures show that spurious oscillation does not occur along the high-gradient boundary layers at these final times. The computed maximum values are also comparable with those obtained from the method of Ref. [13]. 4 Conclusions This paper presents an explicit finite volume element method for analyzing the reaction diffusion equation on twodimensional domain. The theoretical formulation of the combined finite volume and finite element method was explained in details. The finite volume method was applied for discretizing the computational domain, while the finite element method was used to estimate the gradient quantities at the cell faces. The numerical test cases have shown that the combined method does not require any special numerical treatment to improve the solution stability. Three examples were used to evaluate the robustness and efficiency of the combined method. These examples show that the combined method can provide high-resolution profiles without spurious oscillation along the high-gradient boundary layers. Acknowledgments The authors are pleased to acknowledge the Thailand Research Fund (TRF) for supporting this research work. References 1. Apel, T., Lube, G.: Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem. Appl. Numer. Math. 26, (1998) 2. Li, J., Navon, I.M.: Uniformly convergent finite element meth-ods for singularly perturbed elliptic boundary value problems. I. Reaction diffusion type. Comput. Math. Appl. 35, (1998) 3. Castaings, W., Navon, I.M.: Mesh refinement strategies for solving singularly perturbed reaction diffusion problems. Comput. Math. Appl. 41, (2001) 4. Lin, B.T., Madden, N.: A finite element analysis of a coupled system of singularly perturbed reaction diffusion equations. Appl. Math. Comput. 148, (2004) 5. El-Salam, M.R.A., Shehata, M.H.: The numerical solution for reaction diffusion combustion with fuel consumption. Appl. Math. Comput. 160, (2005) 6. Xu, R.: A reaction diffusion predator prey model with stage structure and nonlocal delay. Appl. Math. Comput. 175, (2006) 7. Heineken, W., Warnecke, G.: Partitioning methods for reaction diffusion problems. Appl. Numer. Math. 56, (2006) 8. Parvazinia, M., Nassehi, V.: Multiscale finite element modeling of diffusion reaction equation using bubble functions with bilinear and triangular elements. Comput. Methods Appl. Mech. Eng. 196, (2007) 9. Moore, P.K.: Solving regularly and singularly perturbed reaction diffusion equations in three space dimensions. J. Comput. Phys. 224, (2007) 10. Roos, H.G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Springer-Verlag, Berlin (1996) 11. Xenophontos, C.: Optimal mesh design for the finite element approximation of reaction diffusion problems. Int. J. Numer. Meth. Eng. 53, (2002)

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