A mixed spectral method for incompressible viscous fluid flow in an infinite strip

A mixed spectral method for incompressible viscous fluid flow in an infinite strip^*

Wang Zhong-Qing; Guo Ben-yu

Department of Mathematics, Shanghai Normal University. Division of Computational Science of E-Institute of Shangai Universities. Guilin Road 100, Shanghai, China, 200234. E-mail: [e-mail]/ [e-mail]

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 Legendre-Legendre 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 Legendre-Legendre 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 [12-15]. 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 multiple-dimensional 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 Legendre-Legendre rational spectral method for the Navier-Stokes 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 free-divergence 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 Navier-Stokes equation. Some authors have used the vorticity-stream 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 Navier-Stokes 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 Navier-Stokes 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 Legendre-Legendre rational spectral scheme for the stream function form of Navier-Stokes 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 Legendre-Legendre 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 Legendre-Legendre Rational Spectral Method

In this section, we first propose the mixed Legendre-Legendre rational approximation, and then construct a mixed scheme for the stream function form of Navier-Stokes 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, semi-norm 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 Sturm-Liouville problem

The corresponding eigenvalues l = l(l+1). They satisfy the following recurrence relations

The set of Legendre polynomials is the complete L²(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) = (1-y)^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 = {x|0 < x < ¥}.

The Legendre rational functions of degree l are defined by

According to [18], R_l(x) are the eigenfunctions of the singular Sturm-Liouville 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 L²(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 ₀₀H²(I). Moveover, (2.10) implies that R_l(x) ® 0 and ¶_xR_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 Legendre-Legendre 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 semi-norm , and norm have the meanings as usual. In particular, we denote by (u,v) and ||v|| the inner product and norm of L²(W).

For any u Î (W), u(x,y) = ¶_xu(x,y) = ¶_yu(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: L²(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 Navier-Stokes equation in numerical simulation of incompressible flow. In order to do this, Guo Ben-yu and coauthors developed the Legendre spectral method for a square and the mixed Legendre-Laguerre spectral method for an infinite strip, see [16, 23, 33]. We now construct the mixed Legendre-Legendre 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), U₀(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 Navier-Stokes equation is as follows,

Let f = -Ñ×F and the operator J(u,v,w) = (Dv,¶_yu¶_x w-¶_x u¶_y w). The weak formulation of (2.15) is to find U Î L^¥(0,T;(W))ÇL²(0,T;(W)) such that

Clearly,

It was shown in [22] that if U₀Î (W) and f Î L²(0,T;H^-2(W)), then(2.16) has a unique solution in L^¥(0,T;(W))ÇL²(0,T;(W)).

The mixed Legendre-Legendre 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,MU₀, U₀ or U₀.

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 non-isotropic 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 H¹(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 Runge-Kutta method of fourth-order, 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 = U₀ 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 Legendre-Legendre 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 semi-norm 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, < c||v||_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 = U₀. 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 G₄(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,MU₀ or U₀, 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 Navier-Stokes 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 free-divergence base functions. Indeed, this is not an easy job for spectral methods. Additionally, we have avoided non-physical boundary conditions on the pressure or the vorticity, which usually creates numerical boundary layers.

In this paper, we use the mixed Legendre-Legendre 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 multi-dimensional 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 Legendre-Legendre rational approximation, which form the mathematical foundation of the spectral method for an infinite strip. We may also consider the mixed Legendre-Legendre rational interpolation which leads to the mixed Legendre-Legendre 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].

Received: 30/VIII/04. Accepted: 05/IV/05.

#614/04.

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*

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 E-institutes of Shanghai Municipal Education Commission, N.E03004, The Special Funds for Major Specialities and The Fund N.04DB15 of Shanghai Education Commission.

Publication Dates

• Publication in this collection
  20 Apr 2006
• Date of issue
  Dec 2005

History

• Accepted
  05 Apr 2005
• Received
  30 Aug 2004