Iterative Methods and High-Order Difference Schemes for 2D Elliptic Problems with Mixed Derivative

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1 Iterative Methods and High-Order Difference Schemes for 2D Elliptic Problems with Mixed Derivative Michel Fournié and Samir Karaa Laboratoire MIP, CNRS UMR 5640, Université Paul Sabatier, 118 route de Narbonne, Toulouse Cedex 4, France Department of Mathematics and Statistics, Sultan Qaboos University, P. O. Box 36, Al-Khod 3, Muscat, Sultanate of Oman 1. Introduction We consider the two-dimensional elliptic equation 2 u x u y 2 + β 2 u x y + c u x + d u = fx, y, x, y Ω, 1 y with Dirichlet boundary conditions on Ω, where Ω is a smooth convex domain in IR 2 consisting of a union of rectangular shapes. The coefficients c, d, and β are assumed constant, and β satisfies the ellipticity condition β 2 < 4. The forcing function fx, y as well as the solution ux, y of the problem are assumed sufficiently smooth in Ω. Efficient numerical solution of Equation 1 plays an important role in many areas especially in computational fluid dynamics. Discretizations of 1 using the traditional 5-point difference schemes: either the central difference scheme CDS or the upwind difference scheme UDS, yield unsatisfactory results. The CDS scheme has a truncation error of order Oh 2 but may produce numerical solutions with nonphysical oscillations for large cell Reynolds numbers. The UDS scheme suppresses these oscillations with large artificial viscosity, and stationary iterative methods for solving the resulting linear system are stable converge for large values of Reynolds numbers. However, it is only of first-order accuracy and requires fine discretizations for satisfactory solution resolution. Fine discretization entails increased computational cost, especially for higher dimensional problems. To obtain satisfactory numerical results with reasonable computational cost, several authors proposed improved finite difference discretization schemes that combine the advantages of the second-order central difference scheme high accuracy and the first-order upwind scheme convergence of stationary iterative methods. Among these is the fourth-order compact scheme FCS due to Gupta, Manohar and Stephenson [2] for convection-diffusion problems β = 0 in two 1

2 2 Michel Fournié and Samir Karaa dimensions. The scheme has a 9-point computational stencil using the eight nearest neighboring points of the reference grid point α 6 α 2 α 5 α 3 α 0 α 1, α 7 α 4 α 8 and has a truncation error of order Oh 4. Numerical experiments in [2] showed that this compact scheme has a good numerical stability for large Reynolds numbers, and SOR iterative methods have been found numerically to converge regardless the magnetude of the Reynolds number. This property is very important for implementing multigtrid method which requires similar discretizations on very course grids [3]. Analytic proofs confirming the convergence of some stationary iterative methods with large Reynolds numbers are given in [4] for convection-diffusion problems with constant coefficients. For problems with variable coefficients, several conditions are formulated in [5, 6], under which the iterative methods converge. Comprehensive studies and useful applications of the fourth-order compact scheme in computational fluid dynamics can be found in at least five recent PhD theses [7]-[11]. The fourth-order compact schemes have also been used in the numerical simulation of incompressible Navier-Stokes equations with good results [1,, 13, 14]. Extension of high-order compact schemes to time-dependent problems are reported in [15, 16, 17]. In this paper, we generalize the 9-point compact scheme to the two-dimensional elliptic problem 1 involving a mixed derivative β 0. After derivation of the high-order compact scheme, we conduct experimental study on the numerical solution of the problem discretized by the new scheme and the traditional secondorder central difference scheme. We study the computed accuracy achieved by each scheme and the performance of some iterative methods for solving linear systems arising from the difference schemes. 2. Derivation of High-Order Scheme The idea behind the derivation of the high-order compact scheme is to operate on the differential equations as an auxiliary relation to obtain finite difference approximations for high-order derivatives in the truncation error. Inclusion of these expressions in a central difference method for Equation 1 increase the order of accuracy, typically to Oh 4 while retaining a compact stencil defined by nodes surrounding a grid point. Introducing a uniform grid with mesh spacing h in both the x- and y-directions, the standard central difference approximation to Equation 1 at grid point i, j is simply δ 2 xu ij δ 2 yu ij + βδ x δ y u ij + cδ x u ij + dδ y u ij τ ij = f ij, 2

3 Iterative Methods and High-Order Difference Schemes 3 where δ x and δx 2 resp. δ y and δy 2 denote the first and second order central difference approximations with respect to x resp. with respect to y. The associated truncation error is given by τ ij = h2 [ 2 c 3 u x 3 + u d 3 y 3 4 u x u y 4 ] ij + βh2 6 [ 4 ] u x 3 y + 4 u x y 3 ij +Oh 4. 3 We now seek second-order approximations to the derivatives in 3. Differentiating Equation 1 with respect to x and y respectively yields 3 u x 3 3 u y 3 = 3 u x y 2 + c 2 u x 2 + d 2 u x y + β 3 u x 2 y f x, 4 = 3 u x 2 y + c 2 u x y + d 2 u y 2 + β 3 u x y 2 f y. 5 Differentiating the equation with respect to x and y we obtain 4 u x 3 y + 4 u x y 3 = β 4 u x 2 y 2 + c 3 u x 2 y + d 3 u x y 2 2 f x y. 6 Notice that all the terms in the right hand sides of 4-6 have compact Oh 2 approximations at node i, j. We have for example 3 [ ] u x 2 y = δxδ 2 y u ij h2 2 5 u ij x 2 y u x 4. y Once again by differentiating twice Equation 1 with respect to x and y respectively, we obtain 4 u x 4 = 4 u x 2 y 2 + u c 3 x 3 + d 3 u x 2 y + β 4 u x 3 y 2 f x 2, 7 4 u y 4 = 4 u x 2 y 2 + c 3 u x y 2 + u d 3 y 3 + β 4 u x y 3 2 f y 2. 8 By adding the two equations, we have 4 u x u y 4 = 2 4 u 3 x 2 y 2 + c u x u x y 2 4 u +β x 3 y + 4 u x y 3 3 u + d x 2 y + 3 u y 3 2 f x 2 2 f y 2, 9 which can be approximated to Oh 2 within the 9-point compact stencil. Substituting Equations 4-9 into Equation 3 and simplifying yields an error term of the form

4 4 Michel Fournié and Samir Karaa [ { τ ij = h2 c 3 u x y 2 + u c 2 x 2 + d 2 u x y + β 3 u x 2 y { } +d 3 u x 2 y + c 2 u x y + u d 2 y 2 + β 3 u x y 2 ] +2 4 u x 2 y 2 c 3 u x y 2 d 3 u x 2 y + βh2 + h2 [ ] β 2 u x 2 y 2 + c 3 u x 2 y + d 3 u x y 2 ij [ c f x d f y + 2 f x f y 2 β 2 f x y ij } ] ij + Oh Now, substituting the central Oh 2 approximations to the derivatives in 10 and inserting in 2 yields the following Oh 4 approximation to c 2 h2 h β2 h 2 h 2 = f ij + h2 δ 2 xu ij δ 2 xδ 2 yu ij βh2 βd 2c d 2 h2 h 2 d δ 2 yu ij + βh2 cβ 2d + c β cdh2 6 δxδ 2 y u ij δ x δ y u ij δ x δyu 2 ij + cδ x u ij + dδ y u ij ]. 11 [ c f x d f y + 2 f x f y 2 β 2 f x y ij The fourth-order compact finite difference formula for the mesh point i, j involves the nearest eight neighboring mesh points: 8 8 α l u l = γ l f l, l=0 l=0

5 Iterative Methods and High-Order Difference Schemes 5 where the coefficients α l, l = 0, 1,..., 8, are obtained from 11 α 0 = 20 + d 2 h 2 2 β 2 + c 2 h 2, α 1 = 1 2 α 2 = 1 2 α 3 = 1 2 α 4 = 1 2 α 5 = ch c 2 h 2 2 β 2 2 hβ d, 8 2 β 2 4 dh + d 2 h 2 2 hcβ, 8 + c 2 h 2 2 β ch + 2 hβ d, 8 + d 2 h hcβ + 4 dh 2 β 2, 2 hβ d 2 ch β 2 6 β + cdh hcβ 2 dh, α 6 = hβ d 2 ch 4 2 β 2 6 β + cdh 2 2 hcβ + 2 dh, α 7 = hβ d + 2 ch β 2 6 β + cdh 2 2 hcβ + 2 dh, α 8 = hβ d + 2 ch 4 2 β 2 6 β + cdh hcβ 2 dh. The coefficents γ l, l = 0,, 8 are γ 0 = 4 h 2, γ 1 = 1 4 h2 2 + ch, γ 2 = 1 4 h2 dh 2, γ 3 = 1 4 h2 ch + 2, γ 4 = 1 4 h2 2 + dh, γ 5 = 1 8 h2 β, γ 6 = 1 8 h2 β, γ 7 = 1 8 h2 β, γ 8 = frac18 h 2 β. 3. Numerical Experiments We conduct numerical experiments to test and compare the computed accuracy achieved by the present fourth-order compact FCS scheme and the central scheme. We also examine the numerical performance of a few iterative methods for solving the resulting linear systems. A similar study has been carried out

6 6 Michel Fournié and Samir Karaa in [18] for convection-diffusion problems. We consider the following two test problems c = Re, Test Problem 1. d = Re, ux, y = sinπx + sinπy + sinπx sinπy. Test Problem 2. c = Re, d = Re, ux, y = [e Re1 x + e Rey 2]/[e Re 1]. The problems are solved on a unit square Ω = 0, 1 0, 1 using a uniform meshsize h in the x- and y-directions. The right-hand side fx, y and the Dirichlet boundary conditions are prescribed to satisfy the given exact solution ux, y. Re is referred to as the Reynolds number. In contrast to Problem 1, the exact solution of Problem 2 is related to the Reynolds number. Even for moderate Re Re 100 the solution is nearly identically zero in Ω except at the boundary layers of thickness O1/Re near x = 0 and y = 1. This example is used to test how well the discretization schemes resolve the thin boundary layers. In our experiments, Re is allowed to vary between 1 and 10 6 and β between -2 and Computed accuracy We first test the computed accuracy that was achieved by each discretization scheme. We select GMRES preconditioned by ILUTτ,s to solve the resulting linear systems in all test runs. The parameters τ and s are selected for each test so that full convergence is reached. The computed error is the maximum absolute error over the discretized grid points, and the accuracy order is obtained by comparing the errors after refining the meshsize, i.e., ln 2 e h=1/64 /e h=1/8. All computer programs were coded in Fortran programming language and were run on a SunBlade100 machine. Test results for Problems 1 and 2, with differents values of h, Re, and β are listed in Tables 1 and 2, respectively. The numerical results show that the magnitude of Re affects the computed accuracy of the discretization schemes inversely, and that β has a little affect. For both test problems with Re < 10 3, the fourth-order compact scheme yields much better solution than the central difference scheme, and both schemes seem to maintain their typical orders of accuracy. For Problem 1 with large Re and without boundary layers, the central difference scheme produces accurate solution and succeeds to maintain its order of accuracy. The fourth-order scheme produces also accurate solution, but fails to preserve its high accuracy especially when Re 10 5, and becomes typically

7 Iterative Methods and High-Order Difference Schemes 7 Table 1. Maximum errors and order of accuracy of different discretization schemes Problem 1. Central Difference Scheme Fourth-Order Scheme Re h 1/64 h 1/8 order h 1/64 h 1/8 order β = β = β = of order 2. In [19], Berger et all. showed that in the 1D case, the approximation error of the fourth-order compact scheme is OReh 4 if Reh < 1 and Oh 2 if Reh > 1. Our numerical results show a similarity. We observe that the approximation error in the 2D case is Oh 2 when Reh 2 > 1. For Problem 2 with thin boundary layers, the central difference scheme behaved badly due to the large numerical oscillations. The fourth-order scheme yields satisfactory solution only for Re For Re 10 3, the error produced by the FCS scheme is small but not satisfactory. A careful examination of the computed solution showed that, in contrast to the CDS case, numerical oscillations are not presented in the solution. For large Re, no scheme yielded acceptable results. This boundary layer problem is common to all uniform discretization schemes. The accuracy degradation is mainly caused by the insufficient number of grid points inside the boundary layers. Special treatments, such

8 8 Michel Fournié and Samir Karaa Table 2. Maximum errors and order of accuracy of different discretization schemes Problem 2. Central Difference Scheme Fourth-Order Scheme Re h 1/64 h 1/8 order h 1/64 h 1/8 order β = β = β = as local adaptive mesh refinement techniques or graded mesh techniques, are then needed to resolve the thin boundary layers. The use of nonuniform grids for convection-diffusion problems with boundary layers have been investigated by Gupta, Manohar and Stephenson [2], and Spotz and Carey [20]. It has been found that the fourth-order accuracy can be recovered by employing the graded mesh techniques. A numerical comparison with the central and upwind schemes in such situations is given in [21]. 3.2 Performance of iterative methods We now examine the numerical performance of a few iterative methods for solving the sparse linear systems arising from the discretization schemes. The

9 Iterative Methods and High-Order Difference Schemes 9 first iterative method to be tested is the line Gauss-Seidel method along the x- axis. The grid points with the same y-index are solved simultaneously. One line Gauss-Seidel iteration involves the solution of N tridiagonal linear subsystems of size N each, where N is the number of interior grid points in each direction. Although stationary iterative methods are seldom used as stand alone solvers, they are usually employed as basic components in building modern iterative methods such as the multigrid methods [22] and the Krylov subspace methods [23]. In such cases, basic iterative methods may be used as smoothers in multigrid methods or as preconditioners in Krylov subspace methods. Understanding the behaviors of these iterative methods for solving sparse linear systems arising from the fourth-order compact discretization is practically interesting. Table 3. Number of line Gauss-Seidel iterations for solving the linear systems discretized by the central and compact difference schemes with h = 1/64 Problem 1. β = 0.2 β = 1 β = 1.8 β = 1 β = 1.8 Re CDS FCS CDS FCS CDS FCS CDS FCS CDS FCS indicates divergence. In Table 3 we list the number of Gauss-Seidel iterations for solving the linear systems arising from Problem 1 discretized by the central and compact difference schemes with h = 1/64 and with different values of Re and β. The experimental data show that Gauss-Seidel with the central scheme diverges for Re 10 3 and for all the values of β listed in the table. Several numerical tests reveal that divergence occurs for all values of β in the interval 2, 2, and similar results has been obtained in the case of Problem 2. Gauss-Seidel method with the fourth-order compact scheme is found to converge in all cases, regardless of the magnetude of Re, but its performance deteriorates as Re increases, as well as for small Re. The Table also reveals that the number of Gauss-Seidel iterations for both schemes are close, which means that the amounts of work in both cases do not differ much. The number of Gauss-Seidel iterations as a function of Re with different values of β is shown in Figure 1 for the two test problems. In summary, the numerical experiments can be explained as that the Gauss-Seidel method is more stable with the compact scheme, slightly affected by the values of β when Re 10 2, but heavily affected

10 10 Michel Fournié and Samir Karaa by the magnitude of Re. Its performance is usually good when Re is between 10 2 and We notice that in the non-elliptic case for example when β = 3, the Gauss-Seidel method does not converge for all Reynolds numbers β =1.8 β =1 β = 1 β = 1.8 Problem β =1.8 β =1 β = 1 β = 1.8 Problem Number of GS iterations Number of GS iterations Re Re Figure 1. Number of line Gauss-Seidel iterations as a function of Re for different values of β, for solving the test problems with the compact scheme, h = 1/64. We next solve the linear systems arising from the discretization schemes using a standard multigrid method. The multigrid method has been shown to be very effective for solving discretized elliptic problems [22]. In the present computations, we use a standard multigrid V-cycle algorithm, with the point Gauss-Seidel method as a smoother. We perform one relaxation on each level before projection and after interpolation. Standard full-weighting and bilinear interpolation are employed as the intergrid transfer operators. Standard coarsening technique is used the meshsize of the coarse grid doubles that of the fine grid and all possible levels are exployed, that is the coarsest level has a meshsize h = 1/2. For a description of the multigrid method with different cycling algorithms, we refer to the books [22] and [24]. We choose the finest grid with h = 1/64, the initial guess is ux, y = 0. The iterations are terminated when the 2-norm of the residual is reduced by 10 10, i.e., r n 2 / r 1 2 < Various studies of the multigrid method for solving convection-diffusion equations discretized by the fourth-order compact scheme can be found in [10, 3, 25]. Convergence results for the multigrid method with different Re and β are presented in Tables 4 and 5 for Problems 1 and 2, respectively. The data show that the multigrid method is more stable with the fourth-order compact scheme.

11 Iterative Methods and High-Order Difference Schemes 11 It always converges when Re 10 3 for all values of β with a few exceptions at Re = 10 2, while it diverges with the central scheme when Re 10 2 in all cases. The use of the red-black Gauss-Seidel as a smoother did not improve the results. We believe that the smoother is not the main factor for divergence with the compact scheme, and that suitable intergrid transfer operators with appropriate scaling of the residual, see for instance [26] and [10], can make the multigrid method converge in all cases with the compact scheme. Table 4. Number of Multigrid iterations for solving Problem 1, h = 1/64. β = 0.2 β = 1 β = 1.8 β = 1 β = 1.8 Re CDS FCS CDS FCS CDS FCS CDS FCS CDS FCS indicates divergence. Table 5. Number of Multigrid iterations for solving Problem 2, h = 1/64. β = 0.2 β = 1 β = 1.8 β = 1 β = 1.8 Re CDS FCS CDS FCS CDS FCS CDS FCS CDS FCS indicates divergence. We finally solve the linear systems arising from the discretization schemes using GMRES [27] preconditioned by the incomplete LU preconditioner ILUTτ,s. In full GMRES implementation the storage requirements grow quadratically with the number of iterations. Hence, in practice it is often necessary to use a restarted version, GMRESm, where m indicates the selected dimension of the Krylov subspace. In our case GMRES is restarted after every 20 iterations. The ILUT preconditioner uses a dual truncation dropping strategy developed in [28]. The amount of fill-ins is controlled by two parameters, a threshold drop tolerance τ and a fill-in number s. All entries with magnetude less than s multiplied by the average value of absolute values of the current row are dropped. In addition,

12 Michel Fournié and Samir Karaa Table 6. Number of preconditioned GMRES20 iterations and the amount of fill-ins in ILUT for the different schemes with h = 1/64 and β = 1. Problem 1 Problem 2 CDS FCS CDS FCS Re s Iter. s Iter. s Iter. s Iter Table 7. Number of preconditioned GMRES20 iterations and the amount of fill-ins in ILUT for the different schemes with h = 1/8 and β = 1. Problem 1 Problem 2 CDS FCS CDS FCS Re s Iter. s Iter. s Iter. s Iter indicates divergence. only the largest s entries in each row of the L and U factors are retained. When testing with this preconditioner, it is convenient to set τ to a fixed value, and to vary s to produce a preconditioner with desired amount of fill-ins [29]. In this way, the amount of storage can be predicted in advance and the trade-off between memory cost and computing time can be balanced more easily. In our computations, the dropping tolerenace τ = 10 4 is fixed, and the number of fill-ins s is initially chosen equal to 5. If convergence is not reached within 200 iterations, then s is increased by 5 each time untill we have convergence. The number of preconditioned GMRES iterations and the corresponding parameter s of fill-ins in ILUT for solving the linear systems with h = 1/64 and h = 1/8 are listed in Tables 6 and 7, respectively. We notice that for Problem

13 Iterative Methods and High-Order Difference Schemes 13 2 with Re = 10 5 and h = 1/8, we decreased the dropping tolerance τ to 10 5 in order to achieve convergence with the central difference scheme. The tables indicate that with the fourth-order scheme a small storage space is needed; s = 5 was always enough except for Problem 2 when Re However, with the central scheme, s exceeds 50 in all cases with Re We also remark that when we have the same value for s, GMRES has the least iteration counts with the fourth-order scheme. Since the setup of the ILUT preconditioner requires a certain amount of preprocessing costs depending in part on τ and s, we compare in Figure 2 the total CPU time in seconds, including the setup of the ILUT preconditioner, for solving Problem 2 with h = 1/8. The figure shows a significant growth of the CPU time in the case of the CDS scheme with large Re, while the CPU time remained less than 9 seconds through the entire simulation with the FCS scheme. We can say that preconditioned GMRES is more robust with the FCS scheme, and more effective in terms of computational cost and storage requirement PGMRES with CDS PGMRES with FCS Total CPU time in seconds Figure 2. Comparison of total CPU time in seconds delievered by preconditioned GMRES for solving Problem 2 with the central and the fourth-order schemes h = 1/8. Re 4. Concluding remarks

14 14 Michel Fournié and Samir Karaa We derived a fourth-order compact finite difference scheme for a 2D elliptic problem with a mixed derivative and constant coefficients. Numerical experiments are conducted to test its high accuracy and compare it with the standard central difference scheme. The numerical performances of a few iterative methods for solving the sparse linear systems arising from the discretization schemes are examined. The Gauss-Seidel and multigrid methods showed good stablility with the compact scheme and preconditioned GMRES was found more effective in terms of computational cost and storage requirement. We finally point out that, in the case where the coefficients in the governing equation 1 are not constant, we can derive a compact fourth-order scheme using techniques different from those presented in Section 2. Acknowledgments : S. Karaa s research was supported by Sultan Qaboos University under Grant IG/SCI/DOMAS/05/10 References 1. M. M. Gupta, R. P. Manohar and J. W. Stephenson, A fourth-order, cost effective and stable finite difference scheme for the convection-diffusion equation, In Numerical Properties and Methodologies in Heat Transfer, Washington, DC, 1983, Hemisphere Publishing Corp, M. M. Gupta, R. P. Manohar and J. W. Stephenson, A single cell high-order scheme for the convection-diffusion equation with variable coefficients, Int. J. Numer. Methods Fluids, , M. M. Gupta, J. Kouatchou and J. Zhang, A compact multigrid solver for convectiondiffusion equations, J. Comput. Phys., , S. Karaa and J. Zhang, Analysis of stationary iterative methods for the discrete convectiondiffusion equation with a 9-point compact scheme, J. Comput. Appl. Math., , No. 2, S. Karaa and J. Zhang, A note on convergence of line iterative methods for a nine-point matrix, Appl. Math. Lett., , No. 4, S. Karaa and J. Zhang, Convergence and performance of iterative methods for solving variable coefficient convection-diffusion equation with a fourth-order compact difference scheme, Comput. Math. Appl., , No. 3-4, J. Kouatchou, High-Order Multigrid Techniques for Partial Differential Equations, Ph.D. thesis, The George Washington University, Washington, DC, M. Li, Numerical Solution for the Incompressible Navier-Stokes Equations, Ph.D. thesis, Simon Frasher University, British Columbia, Canada, W. F. Spotz, High-Order Compact Finite Difference Schemes for Computational Mechanics, Ph.D. thesis, University of Texas at Austin, Austin, TX, J. Zhang, Multigrid Acceleration Techniques and Applications to the Numerical Solution of Partial Differential Equations, Ph.D. thesis, The George Washington University, Washington, DC, M. Fournié, Construction et analyse de schémas compacts d ordre élevé pour des problémes fortement convectifs. Applications à la simulation de semi-conducteurs, Ph.D. thesis, Paul Sabatier University, Toulouse, France, 1999.

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