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Computational & Applied Mathematics
Print version ISSN 22383603Online version ISSN 18070302
Comput. Appl. Math. vol.24 no.3 Petrópolis Sept./Dec. 2005
http://dx.doi.org/10.1590/S010182052005000300002
A mixed spectral method for incompressible viscous fluid flow in an infinite strip^{*}
Wang ZhongQing; Guo Benyu
Department of Mathematics, Shanghai Normal University. Division of Computational Science of EInstitute of Shangai Universities. Guilin Road 100, Shanghai, China, 200234. Email: zqwang@shnu.edu.cn/ byguo@shnu.edu.cn
ABSTRACT
This paper considers the numerical simulation of incompressible viscous fluid flow in an infinite strip. A mixed spectral method is proposed using the Legendre approximation in one direction and the Legendre rational approximation in another direction. Numerical results demonstrate the efficiency of this approach. Some results on the mixed LegendreLegendre rational approximation are established, from which the stability and convergence of proposed method follow.
Mathematical subject classification: 65M70, 35Q30, 41A10, 41A20.
Key words: incompressible viscous fluid flow, mixed LegendreLegendre rational spectral method, infinite strip.
1 Introduction
Spectral methods have been used successfully for numerical solutions of differential equations, due to their high accuracy, see, e.g., [2, 5, 6, 7, 9, 10]. The usual spectral methods are available only for bounded domains. However it is also important to consider spectral methods for unbounded domains. Recently, some spectral methods for unbounded domains were proposed, for instance, the Hermite and Laguerre spectral methods, see [8, 11, 17, 23, 26, 29]. By using these methods, we could approximate various differential equations directly. Indeed the weight functions e^{x} and used in these approximations are too strong for some practical problems. We may also reformulate original problems in unbounded domains to singular problems in bounded domains by variable transformations, and then solve the resulting problems by the Jacobi spectral method, see [1215]. In this case, we can use more suitable weight functions and obtain reasonable numerical results oftentimes. However, it is not easy to generalize this approach to multipledimensional problems. Another effective method is based on rational approximations, see, e.g., [3, 4, 18, 19, 20, 32]. So far, all of existing work is only for differential equations of second order.
This paper is devoted to the mixed LegendreLegendre rational spectral method for the NavierStokes equation in an infinite strip. As is well known, this equation plays an important role in studying incompressible viscous fluid flow, see [25, 31]. We usually consider the primitive equation with the velocity u and the pressure p. It is difficult to construct the base functions with freedivergence in spectral methods, and impossible to deal with the boundary values of pressure exactly. Therefore, it seems reasonable to construct numerical schemes based on certain alternative formulations of the NavierStokes equation. Some authors have used the vorticitystream function form, see [25, 27]. However, there is no physical boundary condition on the vorticity. This fact always brings troubles in actual computation. Another way is to consider the stream function form as in [16, 23], in which the incompressibility is fulfilled automatically and the pressure does no longer appear. Moreover, it keeps the physical boundary conditions on the stream function. Thus this form is more appropriate for numerical simulation.
In this work, we shall approximate the stream function form of the NavierStokes equation in an infinite strip by using the Legendre approximation in one direction, and the Legendre rational approximation in another direction. This method has several advantages. Firstly, unlike the Jacobi approximation, we approximate the NavierStokes equation directly. Next, we can use the existing code of the Legendre approximation and so save a lot of work. Thirdly, we use the orthogonal approximation with the Legendre weight function as in the original problem, and so the numerical solution has some conservation properties as in the continuous case. This feature also simplifies actual computation and theoretical analysis.
This paper is organized as follows. In the next section, we propose the mixed LegendreLegendre rational spectral scheme for the stream function form of NavierStokes equation, and present the main results on its stability and convergence. We also present some numerical results demonstrating the spectral accuracy of this method in the spatial variables. In section 3, we first establish some basic results on the mixed LegendreLegendre rational approximation, which plays important role in numerical analysis of the related mixed spectral methods for differential equations of fourth order in an infinite strip. Then we prove the stability and convergence of the proposed scheme. The final section gives some concluding remarks.
2 Mixed LegendreLegendre Rational Spectral Method
In this section, we first propose the mixed LegendreLegendre rational approximation, and then construct a mixed scheme for the stream function form of NavierStokes equation. We state the results on the stability and convergence of the proposed scheme. We also present some numerical results showing the efficiency of this new approach.
2.1 Mixed orthogonal approximation
We first recall the Legendre approximation. Let I = { y  y < 1 } and c(y) be a certain weight function in the usual sense. Denote by the set of all nonnegative integers. For any r Î , we define the weighted Sobolev space (I) in the usual way, and denote its inner product, seminorm and norm by _{} and _{}, respectively. In particular, (I) = (I), _{} and _{}. For any r > 0, we define (I) and its norm by interpolation as in [1]. The space (I) stands for the closure in (I) of the set (I) consisting of all infinitely differentiable functions with compact support in I. When c(y) º 1, we c in the notations as usual.
The Legendre polynomials L_{l}(y) are the eigenfunctions of the singular SturmLiouville problem
The corresponding eigenvalues l = l(l+1). They satisfy the following recurrence relations
The set of Legendre polynomials is the complete L^{2}(I)orthogonal system, namely,
where d_{l,m} is the Kronecker symbol. By virtue of (2.1) and (2.4),
For any N Î , _{N} stands for the set of all algebraic polynomials of degree at most N. Moreover,
In actual computation and numerical analysis, we need two specific Jacobi orthogonal projections. Let c^{(a,b)}(y) = (1y)^{a}(1+y)^{b},a,b> 1 and
The orthogonal projection _{00} : (I) ® _{00}_{N} is defined by
We also define the orthogonal projection : (I) ® , by
We now turn to the Legendre rational approximation. Let
L = {x0 < x < ¥}.
The Legendre rational functions of degree l are defined by
According to [18], R_{l}(x) are the eigenfunctions of the singular SturmLiouville problem
with the corresponding eigenvalues l_{l} = l(l+1). They satisfy the recurrence relations
and
It can be shown that
The set of Legendre rational functions is the complete L^{2}(L)orthogonal system, i.e.,
For any N Î , we set
In order to provide a reasonable algorithm and analyze its convergence properly, we need a specific mapping. To this end, for any v Î (L), let
A simple calculation shows u^{*}(y) Î _{00}H^{2}(I). Moveover, (2.10) implies that R_{l}(x) ® 0 and ¶_{x}R_{l}(x) ® 0, as x ® ¥ . Therefore, by the definition of _{00} and the properties of the Legendre polynomials, we can verify that
Accordingly, we define the mapping : (L) ® by
We now introduce the mixed LegendreLegendre rational approximation. Let W = L × I with the boundary ¶W = {(x,y)  x = 0 or y = ±1}. The spaces H^{r}(W) and (W) with the seminorm , and norm have the meanings as usual. In particular, we denote by (u,v) and v the inner product and norm of L^{2}(W).
For any u Î (W), u(x,y) = ¶_{x}u(x,y) = ¶_{y}u(x,y) = 0 on ¶W. Therefore, we can use the Poincare inequality in one dimension to derive that
Moreover, by integration by parts, we assert that for u Î (W), Du^{2} ~ .
The previous statements tell us that we may take (Du,Dv) as the inner product of (W).
For any
The orthogonal projection P_{N,M}: L^{2}(W) ® V_{N,M} is defined by
The orthogonal projection : (W) ® is defined by
In actual computation and numerical analysis of the mixed spectral methodfor an infinite strip, we shall also use the mapping : (W) ® , defined by
2.2 Mixed spectral scheme
As discussed in Section 1, it is reasonable to use the stream function form of NavierStokes equation in numerical simulation of incompressible flow. In order to do this, Guo Benyu and coauthors developed the Legendre spectral method for a square and the mixed LegendreLaguerre spectral method for an infinite strip, see [16, 23, 33]. We now construct the mixed LegendreLegendre rational spectral method for an infinite strip, which has several advantages in actual computation and theoretical analysis, as described in Section 1.
We denote by U(x,y,t), U_{0}(x,y), n and F(x,y,t) the stream function, the initial state, the kinetic viscosity and the body force, respectively. The stream function form of NavierStokes equation is as follows,
Let f = Ñ×F and the operator J(u,v,w) = (Dv,¶_{y}u¶_{x} w¶_{x} u¶_{y} w). The weak formulation of (2.15) is to find U Î L^{¥}(0,T;(W))ÇL^{2}(0,T;(W)) such that
Clearly,
It was shown in [22] that if U_{0} Î (W) and f Î L^{2}(0,T;H^{2}(W)), then(2.16) has a unique solution in L^{¥}(0,T;(W))ÇL^{2}(0,T;(W)).
The mixed LegendreLegendre rational spectral scheme for (2.16) is to find u_{N,M}(t) Î such that
We can take u_{0,N,M} = P_{N,M}U_{0}, U_{0} or U_{0}.
In the sequel, we denote by c a generic positive constant independent of any function and N, M.
By taking f = 2u_{N,M}(t) in (2.18) and using (2.17), we obtain that
Hence,
2.3 The stability and convergence
In this subsection, we state the results on the stability and convergence of scheme (2.18), which will be proved in section 3.
Theorem 2.1. Assume that u_{0}_{,N,M} and f have the errors _{0}_{,N,M} and , respectively, which induce the error of u_{N,M}, denoted by _{N,M}. Then there exists a positive constant d^{*} depending only on , n and T such that
In order to describe the numerical accuracy properly, we introduce the space (L). For any integer r > 0, its norm is given by
For any r > 0, we define the space (L) by space interpolation. We also introduce the nonisotropic space
equipped with the norms
Furthermore, we define the space M^{r,s}(W). For any integer r,s > 2, its norm is given by
For any r,s > 2, we define the space M^{r,s}(W) by space interpolation. Especially,
For the sake of simplicity, we also denote the norms _{(W)} and _{(W)} by and , respectively.
Theorem 2.2. Let U and u_{N,M} be the solutions of (2.16) and (2.18), respectively. If for integers r,s > 2, U Î H^{1}(0,T;M^{r,s}(W)), then for 0 < t < T,
where d_{*} is a positive constant depending only on n, W, T and the norms ofU in the mentioned spaces.
2.4 Numerical results
We first choose the base functions of suitably. As in [28], let
and
Obviously,
Therefore, we take the base functions of as
The numerical solution is expanded as
Inserting the above expression into (2.18), we obtain a system of ordinary differential equation with unknown function _{N,M}(t). For temporal discretization, we use the standard RungeKutta method of fourthorder, with step size t.
Next, for description of numerical errors, let z_{N,j} and s_{M,k} the distinct roots of R_{N}_{+1}(x) and L_{M}_{+1}(y), respectively. The corresponding weights are denoted by w_{N,j} and r_{M,k}, see [2, 18]. The error
Take the test function
We use (2.18) with u_{0,N,M} = U_{0} to solve (2.16) numerically.
In Table 1, we present the error E_{N,M}(t) at t = 1 for various values of N, M and t. Clearly, the proposed scheme (2.18) provides very accurate numerical solution even for small n and moderate values of N, M and t. They also demonstrate that the error decays fast as N and M increase and t decreases. This coincides well with theoretical analysis.
In Table 2, we present the error E_{N,M}(t) at various values of t. Clearly, the calculation is quite stable.
Table 3 is for the error E_{N,M}(t) at t = 1 with various values of h. It indicates that the errors decay fast as h increases. In fact, the exact solution is smoother for larger h. Therefore, as predicted by Theorem 2.2, the numerical result is more accurate for smoother solution.
In Table 4, we present the error E_{N,M}(t) at t = 1 for various values of n. We find that scheme (2.18) is very efficient even for very small n.
To compare our results with the results in [23], we take the test function
which corresponds to the function W(x,y,t) in (3.1) of [23]. In Table 5, we present the error E_{N,M}(t) of scheme (2.18) at t = 1 and the corresponding results in Table 1 of [23]. Clearly, the proposed scheme (2.18) provides more accurate numerical results for larger values of N, M.
It is noted that the values of N and M in Table 5 correspond to the values of N+4 and M+4 in Table 1 of [23], respectively.
3 Analysis of stability and convergence
In this section, we first establish some basic results on the mixed LegendreLegendre rational approximation which form the mathematical foundation of the related spectral methods for various differential equations in an infinite strip. Then we use these results to prove the stability and convergence of scheme (2.18), stated in Theorems 2.1 and 2.2.
3.1 Some approximation results
Let a,b,g,d,s,l > 1, and introduce the space _{,b,g,d,s,l}(I), 0 <m < 2. For m = 0,_{,b,g,d,s,l}(I) = _{(s,l)}(I). For m = 2,
with the norm
For 0 < m < 2, the space _{,b,g,d,s,l}(I) is defined by the space interpolation as in [1]. Its norm is denoted by u_{m,a,b,g,d,s,l,I}.
For description of approximation results, we also define the space _{,*}(I). For any integer r > 2, its norm and seminorm are given by
For any r > 2, we define the space _{,*}(I) by space interpolation.
Lemma 3.1. If a < min(g+2,s+4), b < 0 and d, l > 0, then for any v Î _{00}(I) Ç_{,*}(I), r Î and r > 2,
If, in addition, a < s+2 and b < l+2, then for 0 < m < 2,
Lemma 3.2. If 1 < a,b < 1, a < g+2 and b < d+2, then for any v Î (I)Ç_{,*}(I), r Î and r > 2,
In particular, for a = b = 0 and 0 < m< 2,
Lemmas 3.1 and 3.2 come from Theorems 2.3 and 2.5 of [21], respectively.
Lemma 3.3. For any v Î (L)Ç(L) and 0 < m < 2 < r,
Proof. By using (2.12), (2.13) and (3.2), a direct calculation leads to that
Moreover, we use induction to show that for k Î ,
c_{j} being certain positive constants. Hence, < cv_{r,A,L}, and so < cN^{r}v_{r,A,L}. For m = 2, we use (2.13) and (3.1) to deduce that
The result with 0 < m < 2 follows from the previous statements and space interpolation.
We now state the main approximation results of this section.
Lemma 3.4. For any v Î M^{r,s}(W)Ç(W) and integers r, s > 2,
Proof. For simplicity of statements, we use the notations
By virtue of (3.4) and (3.5), we deduce that
Similarly
By integration by parts,
Thus, by using (3.7)(3.9) and the Poincare inequality (2.14), we obtain the desired result (3.6).
Lemma 3.5. For any v Î M^{r,s}(W)Ç(W) and integers r, s > 2,
Proof. By projection theorem,
Taking f = v, we have from (3.6) and the Poincare inequality (2.14) that
3.2 Proofs of stability and convergence
We are now in position of proving Theorems 2.1 and 2.2. We shall use two embedding inequalities. In fact, for any u,v,w Î (W) (see [22]),
We first prove Theorem 2.1. According to (2.18), the error _{N,M} satisfies the following equation,
Take f = 2_{N,M} in (3.13). It follows from (2.17) that
Thanks to (2.17), (3.12) and the Cauchy inequality,
By the Poincare inequality (2.14),
Consequently, (3.14) reads
Finally, integrating (3.15) with respect to t and using the Gronwall inequality, we reach the desired result in Theorem 2.1.
We next prove Theorem 2.2. For simplicity, we focus on the case u_{0,N,M} = U_{0}. Let U_{N,M} = U. We have from (2.16) that
where
Further, let _{N,M} = u_{N,M}U_{N,M}. Then subtracting (3.16) from (2.18) yields that
where G_{4}(f,t) = J(U_{N,M}(t),_{N,M}(t),f). In addition, _{N,M}(0) = 0. Take f = 2_{N,M} in (3.17). Then we use (2.17) to deduce that
Next, we estimate G_{j}(_{N,M},t), 1 < j < 4. By the Cauchy inequality and integration by parts, we deduce that
By using (3.11) and the Cauchy inequality,
Similarly,
Also, using (3.12) gives that
Furthermore, due to (3.10), we assert that
Substituting (3.19)(3.25) into (3.18), we obtain that
where
Integrating the above with respect to t and using the Gronwall inequality, we obtain that
This leads to the desired result in Theorem 2.2.
Remark 3.1. If we take u_{0,N,M} = P_{N,M}U_{0} or U_{0}, then we can follow the same line of proof as in the above to obtain similar results.
4 Concluding remarks
In this paper, we consider numerical simulation of the NavierStokes equation which plays an important role in studying incompressible viscous fluid flow. Since we use a numerical algorithm based on the stream function form, the numerical solution fulfills the incompressibility automatically. Moreover, it keeps the physical boundary condition on the stream function. Therefore, we do not need to construct freedivergence base functions. Indeed, this is not an easy job for spectral methods. Additionally, we have avoided nonphysical boundary conditions on the pressure or the vorticity, which usually creates numerical boundary layers.
In this paper, we use the mixed LegendreLegendre rational spectral method for the incompressible fluid flow in an infinite strip. The main characters are the followings:

We approximate the stream function form directly. This fact simplifies actual computation and numerical analysis. It is also easier to generalize the proposed method to multidimensional problems.

We take the Legendre rational functions as the base functions, which are mutually orthogonal with the same weight function as in the continuous version. Thus, the corresponding numerical solution keeps the same conservation as the exact solution. This feature leads to more appropriate numerical results, and simplifies computation and theoretical analysis.

Since the base functions are derived from the Legendre polynomials, we can use the existing code for the Legendre approximation with a slight modification, and so save a lot of work. In particular, we can use Fast Legendre Transformation.

Due to the orthogonality of Legendre polynomials and Legendre rational functions, we provide simple implementation for this mixed spectral method.

Benefiting from the rapid convergence of Legendre and Legendre rational approximations, we obtain very accurate numerical results even for small nodes N and M. The numerical experiments demonstrate the high accuracy of the proposed method.
In this paper, we establish some basic results on the mixed LegendreLegendre rational approximation, which form the mathematical foundation of the spectral method for an infinite strip. We may also consider the mixed LegendreLegendre rational interpolation which leads to the mixed LegendreLegendre rational pseudospectral method for an infinite strip. Clearly this is preferable for actual computations.
In this paper, we use the base functions R_{l}(x) = . But we may apply the scaling base functions
In this case, the adjustable parameter b will offer great flexibility for matching asymptotic behaviors of the exact solutions at infinity. Furthermore, we could also consider Legendre irrational approximation or other mapped Legendre rational approximation, so that the numerical solutions fit the exact solutions more precisely in the region where the exact solutions vary rapidly. We note that the mapped Legendre approximation can also be used for bounded domains, see [30].
REFERENCES
[1] J. Bergh and J.Löfström, Interpolation Spaces, An Introduction, SpringerVerlag, Berlin (1976). [ Links ]
[2] C. Bernardi and Y. Maday, Spectral Methods, in Handbook of Numerical Analysis, Vol. 5, Techniques of Scientific Computing, 209486, edited by P.G. Ciarlet and J.L. Lions, Elsevier, Amsterdam (1997). [ Links ]
[3] J.P. Boyd, Spectral method using rational basis functions on an infinite interval, J. Comp. Phys., 69 (1987), 112142. [ Links ]
[4] J.P. Boyd, Orthogonal rational functions on a semiinfinite interval, J. Comp. Phys., 70(1987), 6388. [ Links ]
[5] J.P. Boyd, Chebyshev and Fourier spectral methods, 2nd edition, Dover Publication Inc., Mineda NY, (2001). [ Links ]
[6]C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin (1988). [ Links ]
[7] D. Funaro, Polynomial Approximations of Differential Equations, SpringerVerlag, Berlin (1992). [ Links ]
[8] D. Funaro and O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite functions, Math. Comp., 57 (1990), 597619. [ Links ]
[9] D. Gottlieb and S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAMCBMS, Philadelphia (1977). [ Links ]
[10] Guo Benyu, Spectral Methods and Their Applications, World Scietific, Singapore (1998). [ Links ]
[11] Guo Benyu, Error estimation of Hermite spectral method for nonlinear partial differential equations, Math. Comp., 68 (1999), 10671078. [ Links ]
[12] Guo Benyu, Gegenbauer approximations and its applications to differential equations on the whole line, J. Math. Anal. and Appl., 226 (1998), 417436. [ Links ]
[13] Guo Benyu, Jacobi spectral approximation and its applications to differential equations on the half line, J. Comp. Math., 18 (2000), 95112. [ Links ]
[14] Guo Benyu, Gegenbauer approximations and its applications to differential equations with rough asymptotic behaviors at infinity, Appl. Numer. Math., 38 (2001), 403425. [ Links ]
[15] Guo Benyu, Jacobi spectral method for differential equations with rough asymptotic behaviors at infinity, J. Comp. Math. Appl., 46 (2003), 95104. [ Links ]
[16] Guo Benyu and He Liping, The fully discrete Legendre spectral approximation of twodimensional unsteady incompressible fluid flow in stream function form, SIAM J. Numer. Anal., 35 (1998), 146176. [ Links ]
[17] Guo Benyu and Jie Shen, LaguerreGalerkin method for nonlinear partial differentialequations on a semiinfinite intervals, Numer. Math., 86 (2000), 635654. [ Links ]
[18] Benyu Guo and Jie Shen, On spectral approximations using modified Legendre rational functions: application to the Kortewegde Vries equation on the half line, Indiana Univ. J. of Math., 50 (2001), 181204. [ Links ]
[19] Benyu Guo, Jie Shen and Zhongqing Wang, A rational approximation and its applications to differential equations on the half line, J. of Sci. Comp., 15 (2000), 117148. [ Links ]
[20] Benyu Guo, Jie Shen and Zhongqing Wang, Chebyshev rational spectral and pseudospectral methods on a semiinfinite interval, Int. J. Numer. Meth. Engng., 53 (2002), 6584. [ Links ]
[21] Guo Benyu, Wang Zhongqing, Wan Zhengsu and Chu Delin, Second order Jacobi approximation with applications to fourthorder differential equations, Appl. Numer. Math., to appear. [ Links ]
[22] Guo Benyu and Xu Chenglong, On twodimensional incompressible fluid flow in an infinite strip, Math. Meth. Appl. Sci., 23 (2000), 16171636. [ Links ]
[23] Guo Benyu and Xu Chenglong, Mixed LaguerreLegendre pseudospectral method for incompressible flow in an infinite strip, Math. Comp., 73 (2004), 95125. [ Links ]
[24] D.B. Haidvogel and T.A. Zang, The accurate solution of Possion's equation by expansion in Chebyshev polynomials, J. Comput. Phys., 30 (1979), 167180. [ Links ]
[25] J.L. Lions, Quelques méthods de résolution des problèmes aux limités non linéaires, Dunod, Paris (1969). [ Links ]
[26] Y. Maday, B. PernaudThomas and H. Vanderen, Reappraisal of Laguerre type spectral methods, La Recherche Aerospatiale, 6 (1985), 1335. [ Links ]
[27] P.J. Roach, Computational fluid dynamics, 2nd ed., Hermosa, Albuquerque, NM, (1976). [ Links ]
[28] Jie Shen, Efficient spectralGalerkin method I. Direct solvers of second and fourth order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 14891505. [ Links ]
[29] Jie Shen, Stable and efficient spectral methods in unbounded domains by using Laguerre functions, SIAM J. Numer. Anal., 38 (2000), 11131133. [ Links ]
[30] Jie Shen and Lilian Wang, Error analysis for mapped Legendre spectral and pseudospectral methods, SIAM J. Numer. Anal., 42 (2004), 326349. [ Links ]
[31] R. Témam, NavierStokes Equation, NorthHolland, Amsterdam (1977). [ Links ]
[32] Wang Zhongqing and Guo Benyu, A rational approximation and its applications to nonlinear partial differential equations on the whole line, J. Math. Anal. Appl., 274 (2002), 374403. [ Links ]
[33] Xu Chenglong and Guo Benyu, Mixed LaguerreLegendre spectral method for incompressible flow in an infinite strip, Advances in Computational Mathematics, 16 (2002), 7796. [ Links ]
Received: 30/VIII/04. Accepted: 05/IV/05.
#614/04.
* The work of these authors is supported in part by NSF of China, N.10471095, The Science Foundation of Shanghai N.04JC14062, The Special Funds for Doctorial Authorities of Chinese Education Ministry N.20040270002, The Einstitutes of Shanghai Municipal Education Commission, N.E03004, The Special Funds for Major Specialities and The Fund N.04DB15 of Shanghai Education Commission.