An Exponential High-Order Compact ADI Method for 3D Unsteady Convection Diffusion Problems

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1 An Exponential High-Order Compact ADI Method for 3D Unsteady Convection Diffusion Problems Yongbin Ge, 1 Zhen F. Tian, 2 Jun Zhang 3 1 Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan, Ningxia , China 2 Department of Mechanics and Engineering Science, Fudan University, Shanghai, , China 3 Department of Computer Science, University of Kentucky, Lexington, Kentucky , USA Received 4 September 2011; accepted 28 November 2011 Published online 5 March 2012 in Wiley Online Library wileyonlinelibrary.com. DOI /num In this article, we develop an exponential high order compact alternating direction implicit EHOC ADI method for solving three dimensional 3D unsteady convection diffusion equations. The method, which requires only a regular seven-point 3D stencil similar to that in the standard second-order methods, is second order accurate in time and fourth-order accurate in space and unconditionally stable. The resulting EHOC ADI scheme in each alternating direction implicit ADI solution step corresponding to a strictly diagonally dominant matrix equation can be solved by the application of the one-dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. Numerical experiments for three test problems are carried out to demonstrate the performance of the present method and to compare it with the classical Douglas Gunn ADI method and the Karaa s high-order compact ADI method Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 29: , 2013 Keywords: ADI method; exponential; high-order compact scheme; stability; three-dimensional unsteady convection diffusion equation I. INTRODUCTION In this article, we consider the solution of the following unsteady three-dimensional 3D convection diffusion equation: u t a 2 u x b 2 u 2 y c 2 u 2 z + p u 2 x + q u y + r u = St, x, y, z z t, x, y, z 0, T ] 1a Correspondence to: Zhen F. Tian, Department of Mechanics and Engineering Science, Fudan University, Shanghai, , China zftian@fudan.edu.cn or z.f.tian@126.com Contract grant sponsor: National Natural Science Foundation of China; contract grant numbers: and Contract grant sponsor: The Key Project of China Ministry of Education; contract grant number: Contract grant sponsor: Fok Ying-Tong Education Foundation of China; contract grant number: Contract grant sponsor: Natural Science Foundation of Ningxia; contract grant number: NZ Wiley Periodicals, Inc.

2 EXPONENTIAL HIGH-ORDER COMPACT ADI 187 with initial conditions and Dirichlet boundary conditions u0, x, y, z = u 0 x, y, z, x, y, z 1b ut, x, y, z = gt, x, y, z, t, x, y, z 0, T ] 1c where is a cubic region in R 3, is the boundary of, 0, T ] is the time interval, and u 0, g, and the source term S are given sufficiently smooth functions. In 1a, p, q, and r are constant convective velocities and a, b, and c are constant, positive diffusion coefficients in the x-, y- and z-direction, respectively. This equation plays a very important role in fluids dynamics and heat transfer as it can be regarded as the simplest prototype equation of the Navier Stokes equations, which are the governing equations of a great number of fluids flow and heat transfer problems. There exist a lot of studies devoted to the numerical approximation to the convection diffusion equations [1 18]. In recent years, high order compact HOC difference methods have been developed by many researchers and used to solve the convection diffusion equations and the Navier Stokes equations [1 7, 9, 12 24]. These methods are shown to be accurate, stable, and cost effective, and treat boundary conditions effectively. Up to now, the HOC difference schemes for solving the steady-state two-dimensional 2D and 3D convection diffusion problems have been studied extensively in the literature. In contrast, the HOC difference methods of unsteady 2D and 3D convection diffusion equations have not been studied to the same extend [1, 9, 12 15, 17]. For the unsteady 2D convection diffusion equation with constant coefficients, Noye and Tan [1] derived a nine-point HOC implicit scheme, which is third order in space and second order in time and has a large stability region. Dehghan and Mohebbi [17] developed a class of high order accurate methods for solving the 2D unsteady convection diffusion equation with variable convection coefficients. The proposed method has fourth-order accuracy in both space and time variables and is unconditionally stable. At each time step, however, a sparse linear system arising from the implicit discretization must be solved. Direct methods based on Gaussian elimination may be too expensive to use for solving such sparse linear system of large size, so, the line Gauss Seidel iterative method was used at each time step. However, they may also be expensive for higher dimensional problems. To achieve higher spatial accuracy and computational cost effectiveness, recently, there has been a renewed interest in the development of HOC ADI methods [9, 12, 13, 15, 25 27]. The ADI methods, which are based on reducing problems in several space variables to collections of one-dimensional 1D problems and only requiring to solve tridiagonal matrices, are highly efficient procedures for the solution of parabolic and hyperbolic initial-boundary value problems. For the 2D unsteady convection diffusion problems, the HOC ADI and the exponential HOC ADI EHOC ADI methods were proposed by Karaa and Zhang [9] and Tian and Ge [15], respectively. These two methods are both second-order accurate in time and fourth-order in space and unconditionally stable. The difference between the HOC ADI scheme and the EHOC ADI scheme is that, for the spatial approximation, the former is polynomial compact difference discretization and the latter is exponential compact difference discretization. On the other hand, the latter [15] produces a strictly diagonally dominant tridiagonal matrix equation that can be solved by using the 1D tridiagonal Thomas algorithm with a considerable saving in computing time. In addition, a distinguishing property of the EHOC ADI method is that it requires significantly fewer number of grid nodes to accurately resolve solution gradients for the convection-dominated problems [15]. For 3D cases, based on the modified equivalent partial differential equation as described by Warming and Hyett [28], Dehghan [10] developed several second order fully explicit and fully

3 188 GE, TIAN, AND ZHANG implicit difference schemes for the 3D convection diffusion equation with constant coefficients. The fully implicit difference schemes are unconditionally stable, where the fully explicit difference schemes are conditionally stable. Based on the idea of the operators splitting method, Wang and Shen [11] proposed two splitting schemes that are fourth-order accurate for spatial diffusion term and second order in time, and a revised ADI scheme that can reach the spatial fourth-order accuracy for convective term. Ge et al. [14] proposed an unconditionally stable three-level implicit HOC difference scheme for the 3D convection diffusion equation with variable convection coefficients. The scheme is fourth-order accurate in space and second order in time. A multigrid algorithm was presented to accelerate the convergence speed of the traditional relaxation methods that are used to treat the implicit difference scheme at each time step. In [12], Karaa presented a polynomial HOC ADI method, which is second order in time and fourth-order in space. It is shown by using a discrete Fourier analysis that the method is unconditionally stable in the diffusion case. However, stability analysis is not given for convection diffusion case. The main aim of this work is to develop a EHOC ADI method for unsteady 3D convection diffusion equation. The present EHOC ADI method is derived by using the fourth-order compact exponential difference operator for the spatial approximation and an exponential difference operator for the temporal approximation, and it is superior to the classical Douglas Gunn ADI method and the Karaa s HOC ADI method [12]. The organization of this article is as follows. Section II presents an EHOC ADI finite difference formula for the 3D unsteady convection diffusion equation; in Section III, the Fourier or von Neumann stability analysis is used to show that the proposed EHOC ADI scheme is unconditionally stable; Section IV presents some numerical experiments to validate the effectiveness of the present methods; finally we draw conclusions in Section V. II. HOC ADI SCHEME Model Eq. 1a, rewritten for the 3D steady-like case, becomes a 2 u x b 2 u 2 y c 2 u 2 z + p u 2 x + q u y + r u = Sx, y, z 2 z where Sx, y, z is a sufficiently smooth function with respect to x, y, and z. To introduce the basic idea, we start from the following 1D steady convection diffusion equation: au xx + pu x = f 3 where a and p are defined as before and f is a sufficiently smooth function of x. Using the approach introduced in [18] see Appendix, it is easy to obtain a three-point fourth-order compact scheme FCS, which can be formulated symbolically as: L 1 x A xu i = f i + O h 4 x 4 in which L x = 1 + α 1 δ x + α 2 δ 2 x, A x = αδ 2 x + pδ x,

4 EXPONENTIAL HIGH-ORDER COMPACT ADI 189 and δ x and δx 2 are the first- and second-order central difference operators, h x is the mesh size in the x-direction, and ph x α = 2 coth phx a α aa α, p = 0 2a, α 1 = p, p = 0 + h2 x p, α 2 = 2 6, p = 0 a, p = 0 0, p = 0 h 2 x 12, p = 0 5 In the same way, we define finite difference operators about y- and z-direction as follow: L y = 1 + β 1 δ y + β 2 δ 2 y, A y = βδ 2 y + qδ y, L z = 1 + γ 1 δ z + γ 2 δ 2 z, A z = γδ 2 z + rδ z, where δ y, δ z, δy 2, and δ2 z are the first- and second-order central difference operators in the y- and z-direction, respectively, and qh y β = 2 coth qhy, q = 0 b β bb β, q = 0 + h2 y 2a, β 1 = q q, β 2 = 2 6, q = 0 b, q = 0 0, q = 0 h 2 y 12, q = 0 6 rh z γ = 2 coth rhz c γ cc γ, r = 0, r = 0 2a, γ 1 = r r, γ 2 = 2 c, r = 0 0, r = 0 + h2 z 6, r = 0 h 2 z 12, r = 0 7 h y and h z are the mesh sizes ofy- and z-direction, respectively. Here, we assume that the equation is approximated on a uniform grid in each direction. Applying the fourth-order compact difference operators L 1 x A x, L 1 y A y, and L 1 z A z to the steady-like 3D convection diffusion Eq. 2, we can obtain the following exponential fourth-order approximation L 1 x A x + L 1 y A y + L 1 z A z uij k = S ij k + Oh 4 8 where Oh 4 denotes the truncated terms of order Oh 4 x + h4 y + h4 z and ijk represents the spatial position of x i, y j, z k. Replacing S by u/ t in Eq. 8, we have u n = L 1 x t A x + L 1 y A y + L 1 z A z u n ij k + Oh4 9 ij k in which u n is the approximate solution at time level t n = n t, n represents the temporal level, and t = t n+1 t n is the time step size. Equation 9 is a fourth-order semidiscrete approximation

5 190 GE, TIAN, AND ZHANG to the unsteady convection diffusion problem 1. In the following, u n will be written in short for u n ij k if there is no confusion about the notations. Using the Taylor series expansion, we have u n = 1 + t + t t 12! t t 2 3! t + u n = exp t u n 10 3 t Combining 10 with 9, a fourth-order difference approximation of Eq. 1 is obtained by exp [ t 2 L 1 x A x + L 1 y u n+1 = exp [ t L 1 x A x + L 1 y A ] y + L 1 z A z u n A ] [ y + L 1 z A z u n+1 = exp t L 1 x 2 A x + L 1 y 11 A ] y + L 1 z A z u n 12 Under the assumptions of the constant convective and diffusion coefficients, the difference operators A x, A y, A z, L x, L y, and L z commute with each other, which yields that t exp 2 L 1 x A x t exp 2 L 1 y A y = exp t exp 2 L 1 z A z u n+1 t 2 L 1 x A x Using the Taylor expansions, it becomes 1 + t = 2 L 1 x A x 1 t 1 + t 2 L 1 x A x 2 L 1 y A y 1 t 1 + t 2 L 1 y A y exp 2 L 1 t z A z 2 L 1 y A y u n+1 exp t 2 L 1 z A z u n 13 1 t 2 L 1 z A z u n + O t 3 + O th 4 14 which is the Crank Nicolson C N time discretization if O t 3 + O th 4 is dropped. When applied to both sides of Eq. 14 with the difference operator L x L y L z,wehave L x + t 2 A x L y + t = 2 A y L x t 2 A x L z + t 2 A z L y t 2 A y u n+1 L z t 2 A z u n + O t 3 + O th 4 15 It is easy to see that the approximation 15 is second-order accurate in time and fourth-order accurate in space and has a compact 27-point stencil. We can obtain the following exponential high-order ADI scheme by dropping the truncation errors O t 3 and O th 4. Introducing two intermediate variables u and u, Eq. 15 can be solved in three steps as L x + t L y + t 2 A x 2 A y L z + t 2 A z u = u = u u n+1 = u L x t 2 A x L y t 2 A y L z t 2 A z u n 16a 16b 16c

6 EXPONENTIAL HIGH-ORDER COMPACT ADI 191 We note that the intermediate values of u and u at the boundary are obtained by u = L z + t g n+1 17 u = L y + t 2 A y 2 A z L z + t 2 A z g n+1 18 For the unsteady 3D convection diffusion equation with a source term, we have the following high-order ADI scheme L x + t 2 A x u = L x t 2 A x L y t 2 A y L z t 2 A z u n L y + t 2 A y L z + t 2 A z + t 2 L xl y L z S n+1 + S n 19a u = u 19b u n+1 = u 19c where S n = Sx i, y j, z k, t n and S n+1 = Sx i, y j, z k, t n+1. Equation 19 is second-order accurate in time and fourth-order accurate in space. III. STABILITY ANALYSIS We need to use von Neumann stability analysis to define the stability limit of the EHOC ADI scheme 16. Let the numerical solution be represented by a Fourier series, whose typical term is u n ij k = ηn exp{iθ x i} exp{iθ y j} exp{iθ z k} 20 where I = 1, η n is the amplitude at time level n, and θ x = σ x h x, θ y = σ y h y, and θ z = σ z h z are phase angles with the wavenumbers σ x, σ y, and σ z in the x-, y- and z-direction, respectively. Substituting the discrete Fourier mode 20 into 15, we obtain the amplification factor Gθ x, θ y, θ z = η n+1 /η n, as follow: where l x θ x is given by with ξ 1 = 1 4α 2 h 2 x Gθ x, θ y, θ z = l x θ x l y θ y l z θ z l x θ x = ξ 1 ξ 2 + Iξ 3 ξ 4 ξ 1 + ξ 2 + Iξ 3 + ξ 4 sin 2 θ x 2, ξ 2 = 2α t sin 2 θ x h 2 x 2, ξ 3 = α 1 sin θ x, h x ξ 4 = p t 2h x sin θ x and the similar expressions for l y θ y and l z θ z may be written in a similar way by replacing x by y and z, p by q and r, α by β and γ in the above expressions, respectively. Using the approach described in [15], it can be proved that l x θ x 1. Similar conclusions can be obtained for l y θ y and l z θ z, that is, l y θ y 1 and l z θ z 1. Thus, the new method is unconditionally stable.

7 192 GE, TIAN, AND ZHANG TABLE I. L 2 error with t = , t = 0.025, and different h, Problem 1. Douglas Gunn ADI scheme Karaa s ADI scheme Present ADI scheme h L 2 error Rate L 2 error Rate L 2 error Rate 1/ / / / IV. NUMERICAL EXPERIMENTS In this section, we present some numerical experiments to illustrate the validity and effectiveness of the proposed EHOC ADI method. We will apply it to three test problems whose exact solutions are given. The numerical results got from the present method will be compared with those from the Douglas Gunn ADI scheme and the Karaa s HOC ADI method [12]. All computations were run on a P4/2.4G private computer using double precision arithmetic. A. Problem 1 Consider a pure diffusion p = q = r = 0 problem in the unit cubic domain 0 x, y, z 1 with diffusion coefficients a = b = c = 1, and so that the exact solution is given by Sx, y, z, t = 2π 2 e π2t sinπx sinπy sinπz ux, y, z, t = e π2t sinπx sinπy sinπz The initial and Dirichlet boundary conditions are directly taken from this exact solution. We use the present EHOC ADI, the Douglas Gunn ADI, and the Karaa s HOC ADI schemes to compute the numerical solutions of this problem. Uniform grids h = h x = h y = h z are used and different mesh sizes are considered. Compared accuracy under the L 2 norm errors of the numerical solution with respect to the exact solution is shown in Tables I III. The rate of convergence is defined by log 2 err1/err2, where err1 and err2 are the L 2 norm errors with the spatial grid sizes h and h/2 or temporal grid sizes t and t/2, respectively. First, we choose a fixed time step t = and halve the spatial grid sizes h from 1/5 to 1/40 in sequence for the verification of fourth-order accuracy in space. L 2 norm errors at t = and the rate of convergence are list in Table I. It is clear that the Karaa s HOC ADI scheme and the present EHOC ADI scheme are fourth-order accurate in space, whereas the Douglas Gunn TABLE II. L 2 error with h = 0.01, t = 0.25, and different t, Problem 1. Douglas Gunn ADI scheme Karaa s ADI scheme Present ADI scheme t L 2 error Rate L 2 error Rate L 2 error Rate 1/ / / /

8 EXPONENTIAL HIGH-ORDER COMPACT ADI 193 TABLE III. L 2 error and CPU time with t = h 2 /3, t = 0.2, Problem 1. Douglas Gunn ADI scheme Karaa s ADI scheme Present ADI scheme h CPU L 2 error Rate CPU L 2 error Rate CPU L 2 error Rate 1/ / / / ADI scheme is second-order accurate in space. If we define error as C 1 h 4 + C 2 t 2, we can get C from the data in Table I. Then, we fix h as 0.01 and halve t in sequence to verify the second-order accuracy in time. L 2 norm errors at t = 0.25 and the rate of convergence are list in Table II. We can see that all three schemes have second-order accuracy in time. Similarly, we can derive C from the data in Table II. To achieve best possible performance, we balance the ratio between h 2 and t by setting C 1 h 4 C 2 t 2. Consequently, t/h 2 C 1 /C 2 1/2 1/3. If we define the ratio as λ = t/h 2, then λ 1/3. Finally, we set λ 1/3 and refine both t and h simultaneously to show that the present method is fourth-order accurate in space and second-order accurate in time. L 2 norm errors at t = 0.2, CPU time, and the rate of convergence are listed in Table III. We can see that both the present EHOC ADI and the Karaa s HOC ADI schemes achieved fourth-order accuracy in space and second-order accuracy in time, whereas the Douglas Gunn ADI scheme achieved just the second-order accuracy in both space and time. The present EHOC ADI scheme provides more accurate solution than the Douglas Gunn ADI scheme or the Karaa s HOC ADI scheme. Furthermore, if we seek a required accuracy, the HOC ADI schemes take far less CPU time by doing computations on coarser grids. For example, the Karaa s HOC ADI scheme and the present EHOC ADI scheme achieve the accuracy of and , with h = 1/10 in s and s, respectively. However, the Douglas Gunn ADI scheme takes s with h = 1/40 to achieve the accuracy of B. Problem 2 Consider a 3D convection and diffusion problem that is defined in the cubic region bounded by 0 x, y, z 2, with an analytical solution given, as in [10, 12], by ux, y, z, t = 4t x pt 0.52 y qt 0.52 exp [ a4t + 1 b4t + 1 ] z rt 0.52 c4t + 1 The Dirichlet boundary and initial conditions are directly taken from this analytical solution. As in [10, 12], we choose the parameters p = q = r = 0.8 and a = b = c = At first, we set h = 0.025, t = 1.25, and change λ from 5 to 40. The L 2 norm errors and the CPU times used for the present EHOC ADI, the Douglas Gunn ADI, and the Karaa s HOC ADI schemes are depicted in Table IV. We notice that for all λ selected, the three schemes are stable. The results show that the present EHOC ADI and the Karaa s HOC ADI schemes provide more accurate solution than Douglas Gunn ADI scheme. As far as CPU time is concerned, the Douglas Gunn ADI scheme needs the least CPU time for each λ. However, if we seek a required accuracy, the HOC ADI schemes can save CPU cost by using bigger time steps bigger λ. For example, the present EHOC ADI scheme achieves the accuracy of with λ = 40 in

9 194 GE, TIAN, AND ZHANG TABLE IV. L 2 error with h = 0.025, t = λh 2, t = 1.25, and different λ, Problem 2. Douglas Gunn ADI scheme Karaa s ADI scheme Present ADI scheme λ L 2 error CPU L 2 error CPU L 2 error CPU s. But the Douglas Gunn ADI scheme takes almost seven times CPU time with λ = 5to achieve the accuracy of Figure 1 gives comparison of solution of Douglas Gunn a, Karaa s [12] b and present c ADI schemes with exact solution on 0 x = y 2, z = 1.2 at h = 0.05, t = It shows that under this conditions, Karaa s and present ADI schemes are both accurate, whereas Douglas Gunn ADI scheme is not accurate. For further comparison, we change the convective coefficients to p = q = 0.8, r = 0.1, and a = b = c = Then we compare the solution of Douglas Gunn a, Karaa s [12] b, and present c ADI schemes with exact solution on 0 x = y 2, z = 0.2 at h = 0.05, t = 1.25 in Fig. 2. We can see that present scheme can still capture very well with the exact solution curve, whereas Karaa s scheme gets a bit bigger amplitude than the original pulse and the distribution is distorted slightly. This fact can also be seen in Fig. 3. C. Problem 3 To further study the superiority of the present ADI method, we consider a 3D steady-state convection diffusion equation in the cubic domain 0 x, y, z 1 with diffusion coefficients a = b = c = 1 and convection coefficients p = Re and q = r = 0. In this case, the convection coefficients are constant but the exact solution exhibits a boundary layer near x = 1. The steepness of the boundary layer depends on the Reynolds number Re. The exact solution is ux, y, z = 1 dxsiny sinz with dx = erex 1 e Re 1. The Dirichlet boundary conditions are directly taken from this exact solution. This problem is used as a test one in Ref. [6], which proposes a FCS and uses a coordinate transformation to remove the boundary layer. For comparison purposes, computations using the present EHOC ADI, Douglas Gunn ADI, and the Karaa s HOC ADI schemes [12] are carried out on uniform grids with a time step t = 0.01 for Re from 1 to Initial guess is zero. The program is terminated when the difference in L norm between u n and u n+1 is less than 10 14, that is u n+1 u n for all grid points, where n and n + 1 are defined as above. Table V shows L 2 norm errorsof the computed solution with respect to the exact solution, the time marching steps for getting steady solutions and the CPU times used. We find that when Re = 1, the HOC ADI and EHOC ADI schemes exhibit very high accuracy, but the HOC ADI scheme needs more time marching steps and more CPU times. With Re increasing to 10, 100, and 1000, the accuracy of Douglas Gunn ADI scheme and HOC ADI scheme deteriorates fast while the EHOC ADI method seems to be more

10 EXPONENTIAL HIGH-ORDER COMPACT ADI 195 FIG. 1. Comparison of Douglas Gunn a, Karaa s [12] b, and present c ADI schemes with exact solution on 0 x = y 2, z = l.2 at h = 0.05, t = 1.25.

11 196 GE, TIAN, AND ZHANG FIG. 2. Comparison of Douglas Gunn a, Karaa s [12] b, and, present c ADI schemes with exact solution on 0 x = y 2, z = 0.2 at h = 0.05, t = 1.25.

12 EXPONENTIAL HIGH-ORDER COMPACT ADI 197 FIG. 3. Contour lines of the pulse: a exact dash line and Douglas Gunn ADI scheme solid line, b exact dash line and Karaa s ADI scheme [12] solid line, and c exact dash line and the present ADI scheme solid line, on z = 0.2 at h = 0.05, t = 1.25.

13 198 GE, TIAN, AND ZHANG FIG. 3. continued and more accurate. From Table V, we can see that three ADI methods get convergent computational results for all Re and the present EHOC ADI method produces amazingly satisfying solution with the increase of Re, whereas the Douglas Gunn ADI scheme and the HOC ADI scheme give poor solutions when Re Numerical solutions computed by the present EHOC ADI, the Douglas Gunn ADI, and the Karaa s HOC ADI schemes for Re = 1, 1000, and 100,000 are also presented in Figs It is seen from Fig. 4 that the three ADI schemes produce very accurate solutions for diffusion-dominated case with Re = 1. However, from Figs. 5 and 6, we can see that the computed solutions for the Douglas Gunn ADI and HOC ADI methods are unacceptable for TABLE V. L 2 errors of three ADI schemes under different Re, with t = 0.01 and h = 1/32, Problem 3. Douglas Gunn ADI scheme Karaa s ADI scheme Present ADI scheme Re Steps CPU L 2 error Steps CPU L 2 error Steps CPU L 2 error

14 EXPONENTIAL HIGH-ORDER COMPACT ADI 199 FIG. 4. Solutions of Problem 3 with Re = 1: a exact, b Douglas Gunn ADI scheme, c Karaa s ADI scheme [12], and d the present ADI scheme. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] convection-dominated cases with Re = 1000, 100,000. In contrast, the EHOC ADI scheme still yields very accurate monotone solutions. Thus, it can be concluded that the HOC ADI method is accurate only when Re is very small while the EHOC ADI scheme achieves very accurate results from low to high Re even though the solution exhibits steep boundary layer. V. CONCLUDING REMARKS In this study, we have developed an EHOC ADI method for the 3D convection diffusion equation. The method is fourth-order in space, second order in time, and proved to be unconditionally stable

15 200 GE, TIAN, AND ZHANG FIG. 5. Solutions of Problem 3 with Re = 1000: a exact, b Douglas Gunn ADI scheme, c Karaa s ADI scheme [12], and d the present ADI scheme. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] by a discrete Fourier analysis. Because it only involves three-point stencil for each 1D operator, it can be inverted by simple tridiagonal Gaussian decomposition and may be solved by applying the 1D tridiagonal Thomas algorithm with a considerable saving in computing time. Furthermore, a distinguishing merit of the present EHOC ADI method is that it requires significantly fewer number of grid nodes to accurately resolve solution gradients for the convection-dominated problem. Numerical experiments are performed to demonstrate its high accuracy and efficiency and to show its superiority over the classical Douglas Gunn ADI scheme and the Karaa s high-order ADI scheme.

16 EXPONENTIAL HIGH-ORDER COMPACT ADI 201 FIG. 6. Solutions of Problem 3 with Re = 100,000: a exact, b Douglas Gunn ADI scheme, c Karaa s ADI scheme [12], and d the present ADI scheme. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] APPENDIX To get a three-point FCS of the following 1D steady convection diffusion equation: auxx + pux = f A1 we consider the following two-point boundary value problem in the sub-domain [xi 1, xi+1 ] i = 1, 2,..., N 1. auxx + pux = f, xi 1 < x < xi+1 A2 uxi 1 = ui 1, uxi+1 = ui+1

17 202 GE, TIAN, AND ZHANG By virtue of the Green function method [29], we obtain the solution of problem A2 xi+1 ux = ϕ 1 xu i 1 + ϕ 2 xu i+1 + Gx, ηf ηdη, A3 x i 1 where the functions ϕ 1 x and ϕ 2 x are the solutions of the following problems, respectively { aϕxx + pϕ x = 0, x i 1 <x<x i+1 ϕx i 1 = 1, ϕx i+1 = 0 A4 and { aϕxx + pϕ x = 0, x i 1 <x<x i+1 ϕx i 1 = 0, ϕx i+1 = 1 A5 which can be explicitly expressed as: ϕ 1 x = 1 e p a x x i+1, ϕ 1 e 2p 2 x = e p a x x i 1 1 a hx e 2p a hx 1 A6 and Gx, η is the Green function of the following problem { auxx + pu x = 0, x i 1 <x<x i+1 ux i 1 = 0, ux i+1 = 0 A7 which can be expressed as [30]: Gx, η = 1 { ϕ1 xϕ 2 η, x i 1 η<x A8 Wη ϕ 1 ηϕ 2 x, x η x i+1 in which, Wη = a e 1 ηxi 1 pdτ xi+1 e 1 ηxi 1 a pdτ dη x i 1 = p a e p a η x i e p a hx e p a hx. A9 From A3 we can obtain the solution of problem A1 xi+1 ux i = ϕ 1 x i u i 1 + ϕ 2 x i u i+1 + Gx i, ηf ηdη A10 x i 1 Using parabolic interpolation method, the source term fxmay be expressed as: where fx= f i + x x i δ x f i + x x i 2 2! δ 2 x f i + x x i 3 f xxx ξ i, ξ i x i 1, x i+1, 3! A11 δ x f i = fx i+1 fx i 1, δ 2 x 2h f i = fx i+1 2fx i + fx i 1. A12 x h 2 x

18 EXPONENTIAL HIGH-ORDER COMPACT ADI 203 For x x i 1, x i+1, i = 1, 2,..., N 1, substituting A11 into A10, combining A8 and rearranging it, we have αδ 2 x u i + pδ x u i = f i + α 1 δ x f i + α 2 δ 2 x f i + Oh 4 x, A13 where δ x and δx 2 defined by δ x u i = u i+1 u i 1 2h x, δ 2 x u i = u i+1 2u i + u i 1 h 2 x A14 are the standard-second order central difference operators, and α = ph x 2 coth phx, α 1 = a α 2a p, α aa α 2 = + h2 x p 2 6. A15 Omitting the truncation error, we obtain the following three-point fourth-order compact finite difference formulation for Eq. A1 αδ 2 x u i + pδ x u i = f i + α 1 δ x f i + α 2 δ 2 x f i. A16 When p = 0, Eq. A1 reduces to au xx = f. A17 Using the fourth-order Padé approximations for the second-order derivative, we get u xxi = δ 2 x 1 + h2 x 12 δ2 x u i + O h 4 x. A18 Substituting A18 into A17 and omitting the truncation error, a three-point fourth-order compact finite difference formulation for Eq. A17 is obtained by aδ 2 x u i = 1 + h2 x 12 δ2 x f i A19 Combining Eqs. A15, A16, and A19, the generalized three-point fourth-order compact finite difference formulation for Eq. A1 can be given by αδ 2 x + pδ xu i = 1 + α 1 δ x + α 2 δ 2 x f i, A20 where ph x α = 2 coth phx a α aa α, p = 0 2a, α 1 = p, p = 0 + h2 x p, α 2 = 2 6, p = 0. a, p = 0 0, p = 0 h 2 x 12, p = 0 A21

19 204 GE, TIAN, AND ZHANG Clearly, Eq. A20 gives rise to a diagonally dominant tridiagonal system of equations and can be formulated symbolically as: in which L 1 x A xu i = f i A22 L x = 1 + α 1 δ x + α 2 δ 2 x, A x = αδ 2 x + pδ x, A23 and δ x and δx 2 are the first and second order central difference operators. The above formulation can be found in Ref. [18] and is repeated here for the sake of completeness. References 1. B. J. Noye and H. H. Tan, Finite difference methods for solving the two-dimensional advection -diffusion equation, Int J Numer Methods Fluids , G. Q. Chen, Z. Gao, and Z. F. Yang, A perturbational h 4 exponential finite difference scheme for the convective diffusion equation, J Comput Phys , J. Zhang, An explicit fourth-order compact finite difference scheme for three dimensional convectiondiffusion equation, Commun Numer Methods Eng , S. M. Choo and S. K. Chung, High-order perturbation-difference scheme for a convection diffusion problem, Comput Methods Appl Mech Eng , J. Zhang, L. X. Ge, and M. M. Gupta, Fourth-order compact difference schemes for 3D convection diffusion equation with boundary layers on nonuniform grid, Neural Parallel Sci Comput , A. C. Radhakrishna Pillai, Fourth-order exponential finite difference methods for boundary value problems of convective diffusion type, Int J Numer Methods Fluids , W. F. Spotz and G. F. Carey, Extension of high-order compact scheme to time-dependent problems, Numer Methods Partial Differential Eq , M. Dehghan, A new ADI technique for two-dimensional parabolic equation with an integral condition, Comput Math Appl , S. Karaa and J. Zhang, High order ADI method for solving unsteady convection-diffusion problems, J Comput Phys , M. Dehghan, Numerical solution of the three-dimensional advection diffusion equation, Appl Math Comput , S. D. Wang and Y. M. Shen, Three high-order splitting schemes for 3D transport equation, Appl Math Mech , S. Karaa, A high-order compact ADI method for solving three-dimensional unsteady convection diffusion problems, Numer Methods Partial Differential Eq , D. You, A high-order pade ADI method for unsteady convection-diffusion equations, J Comput Phys , Y. B. Ge, Z. F. Tian, and W. Q. Wu, Multigrid method based on high accuracy full implicit scheme of 3D convection diffusion equation, Chin J Comput Mech , , in Chinese. 15. Z. F. Tian and Y. B. Ge, A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems, J Comput Appl Math , Z. F. Tian and S. Q. Dai, High-order exponential finite difference methods for convection diffusion type problems, J Comput Phys ,

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