AN EFFICIENT IMPLICIT MESH-FREE METHOD TO SOLVE TWO ...

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International Journal of Modern Physics C Vol. 16, No. 3 (2005) 439–454 c World Scientific Publishing Company

AN EFFICIENT IMPLICIT MESH-FREE METHOD TO SOLVE TWO-DIMENSIONAL COMPRESSIBLE EULER EQUATIONS

H. Q. CHEN Department of Aerodynamics Nanjing University of Aeronautics and Astronautics 29 Yudao Jie, Nanjing 210016, P. R. China C. SHU∗ Department of Mechanical Engineering National University of Singapore 10 Kent Ridge Crescent, Singapore 117576 [email protected]

Received 18 June 2004 Revised 18 October 2004 Local radial basis function-based differential quadrature (RBF-DQ) method is a natural mesh-free approach, in which any derivative of a function at a point is approximated by a weighted linear sum of functional values at its surrounding scattered points. In this paper, the weighting coefficients in the spatial derivative approximation of the Euler equation are determined by using a weighted least-square procedure in the frame of RBFs, which enhances the flexibility of distributing points in the computational domain. An upwind method is further introduced to cope with discontinuities by using Roe’s approximate Riemann solver for estimation of the inviscid flux on the virtual mid-point between the reference knot and its surrounding knot. The lower–upper symmetric Gauss– Seidel (LU-SGS) algorithm, which is implemented in a matrix-free form like the one used in the finite-volume method, is introduced in the work to speed up the convergence. The proposed approach is validated by its application to simulate transonic flows over a NACA 0012 airfoil. It was found that the present mesh-free results agree very well with available data in the literature, and the implicit LU-SGS algorithm can greatly save the computational time as compared with explicit time marching methods. Keywords: RBF-DQ method; Euler equations; LU-SGS compressible flows; least-square approach; implicit method.

algorithm;

transonic

1. Introduction The development of numerical algorithms for the solution of partial differential equations (PDEs) includes the discretization of governing equations in a mesh or in ∗ Corresponding

author. 439

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a set of scattering points. Mesh-based discretization is usually based on the structured mesh or unstructured mesh. The conventional finite difference scheme is based on the structured mesh. The structured mesh is easy to generate but it is not very flexible to treat complex geometry. To be flexible to accommodate geometric complexity, the unstructured mesh1 is usually adopted. However, more computational effort would be paid for generating complicated unstructured meshes and developing related, suitable flow solvers as compared with that used in the structured mesh.2 To eliminate completely the connectivity limitation of the mesh with pointbased discretization, a class of methods, namely “meshless or mesh-free method”, was developed, which behaves naturally and more flexibly to cope with the flow past any complicated geometry due to the following facts. The spatial derivative approximation at any given point by meshless methods depends only on the information of its surrounding points which do not need to form a mesh. The point distribution in the computational domain can be made by using any existing means like the ones used in structured, unstructured, and Cartesian mesh generators. The interesting features motivated many researchers to study this issue and various mesh-free approaches3– 13 have been proposed. Among them, smoothed particle hydrodynamics method,3 the diffuse element method,4 the element free Galerkin method,5 the reproducing kernel particle method,6 the hp-clouds method,7 the finite point method,8 and the general or upwind finite difference method9,10 are the better known methods. In aerodynamics, the most notable work was done by Batina,11 who developed an explicit solver based on the centered scheme with artificial dissipation for solving compressible flows with shocks. An implicit solver was later developed by Morinishi12,13 using “mid-point upwind” and weighted least-squares. Most of above mesh-free methods use a least square technique or its equivalence to construct the mesh-free interpolant for approximating spatial derivatives. The interpolant determined by the so-called radial basis functions (RBFs) forms another group of mesh-free methods for solving PDEs. Initially, RBFs were developed for multivariate data and function interpolation. The method used for solving PDEs was first explored by Kansa14,15 and later contributed by Fornberg,16 Hon and Wu,17 Chen et al.,18 Fasshauer,19 Chen,20 Chen and Tanaka.21,22 It should be noted that most of the above works related to the application of RBFs for the numerical solution of PDEs are actually based on the function approximation instead of derivative approximation. In other words, these works substitute the expression of function approximation by RBFs into a PDE, and then change the dependent variables into the coefficients of function approximation. The derivatives of a PDE can be directly approximated due to the recent work of Shu et al.23 They developed a new method, namely the RBF-based differential quadrature (RBF-DQ) method, which combines the mesh-free nature of RBFs with the derivative approximation of differential quadrature (DQ) method.24 It has been demonstrated that RBF-DQ can be applied to solve any incompressible flow problem. In this paper, we will develop an implicit solver for treating compressible transonic flows in the frame of RBF-DQ method. In Sec. 3, a description of RBF-DQ

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method is presented briefly, followed by an analysis of problems occurred for treating compressible flows in Sec. 4. The two treatments associated with coefficient redistribution are addressed in Sec. 5. Following the idea of evaluating the inviscid flux on the virtual mid-points, an upwind mechanism is further introduced by using Roe’s approximate Riemann solver in Sec. 6. Then the lower–upper symmetric Gauss–Seidel (LU-SGS) algorithm is implemented in Sec. 7 by a matrix-free form like the one used in the finite-volume method. The numerical results of transonic flows over a NACA 0012 airfoil are presented in Sec. 8. They compare very well with available data in the literature. The performance of the proposed least square local RBF-DQ method is well tested. 2. Governing Equations The Euler equations governing compressible inviscid flows can be expressed in dimensionless, conservative form as: ∂W ∂E(W) ∂F(W) + + = 0, ∂t ∂x ∂y

(1)

where the unknown vector W, inviscid or convective flux vectors E and F are defined by: W = (ρ, ρu, ρν, e)T ,

(2)

E(W) = (ρu, ρu2 + p, ρuν, (e + p)u)T ,

(3)

F(W) = (ρν, ρuν, ρν 2 + p, (e + p)ν)T .

(4)

Here ρ, p, and e denote the density, pressure, and specific total energy of the fluid, respectively. The above system can be closed up by using the following equation of state, 1 (5) p = (γ − 1) e − ρ(u2 + ν 2 ) , 2 which is valid for perfect gas, where γ is the ratio of the specific heats. The steady solution of Eq. (1) can be obtained by marching in time until the residuals are less than the given threshold. We will address this issue in the following sections. 3. Local RBF-DQ Method The local RBF-DQ method permits the simplicity of the traditional finite difference method and has shown the ability to deal with both linear and nonlinear incompressible flow problems. Among the RBF-DQ method, MQ-DQ (multiquadrics (MQ)-based differential quadrature (DQ), MQ is a kind of RBFs) is the most popular version. The fundamental issues of local MQ-DQ method can be found in Ref. 23 for details. The procedure is briefly presented here for the sake of completeness.

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Reference knot Supporting knots Nonsupporting knots

Fig. 1.

A set of supporting knots around a reference knot.

Following the work of Shu et al.,23 the nth-order derivative of a function f (x, y) with respect to x, and its mth-order derivative with respect to y, at (xi , yi ) can be approximated by local MQ-DQ method as: N ∂ n f (n) = βik fk , (6) ∂xn i

k=1

N ∂ m f (m) = β¯ik fk , ∂y m i

(7)

k=1

(n) (m) where N is the number of knots used in the supporting region, βik , β¯ik are the DQ weighting coefficients in the x and y directions, and fk is scattered function value at point (xk , yk ). The determination of weighting coefficients is based on the analysis of function approximation and the analysis of linear vector space. In the local MQ-DQ method, MQ approximation is only applied locally. As shown in Fig. 1, at a knot i, there is a supporting region represented by a circle forming a set, namely S(i), of N knots uniformly or randomly distributed. The function in this region can be locally approximated by MQ RBFs as:

f (x, y) =

N

λj

j=1

(x − xj )2 + (y − yj )2 + c2j + λN +1 ,

(8)

where λj is a constant to be determined, cj is the shape parameter to be given by the user. According to the work of Yoon,25 Eq. (8) is subjected to the following condition N j=1

λj = 0 ⇒ λi = −

N j=1,j=i

λj .

(9)

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Substituting Eq. (9) into Eq. (8) gives N

f (x, y) =

λj gj (x, y) + λN +1 ,

(10)

j=1,j=i

where gj (x, y) =

(x − xj )2 + (y − yj )2 + c2j − (x − xi )2 + (y − yi )2 + c2i .

(11)

The number of unknowns in Eq. (10) is N . As no confusion rises, λN +1 can be replaced by λi , and Eq. (10) can be written as: f (x, y) =

N

λj gj (x, y) + λi

(12)

j=1,j=i

It is easy to see that f (x, y) in Eq. (12) constitutes a N -dimensional linear vector space VN with respect to the operation of addition and multiplication. From the concept of linear independence, the bases of a vector space can be considered as linearly independent subset that spans the entire space. In the space VN , one set of base vectors is gi (x, y) = 1, and gj (x, y), j = 1, . . . , N but j = i given by Eq. (11). From the property of a linear vector space, if all the base functions satisfy the linear equation (6) or (7), so does any function in the space VN represented by Eq. (12). Thus, when the weighting coefficients of DQ approximation are determined by all the base functions, they can be used to discretize the derivatives in a partial differential equation. Substituting all the base functions into Eq. (6), we can obtain 0=

N k=1

(n)

βik ,

(13a)

N

(n) ∂ n gj (xi , yi ) = βik gj (xk , yk ) , ∂xn

j = 1, 2, . . . , N ,

but j = i ,

(13b)

k=1

where the derivatives of base function gj (x, y) can be easily obtained. For example, the first-order derivative of gj (x, y) with respect to x can be written as: ∂gj (x, y) x − xj x − xi = − . 2 ∂x (x − xi ) + (y − yi )2 + c2i (x − xj )2 + (y − yj )2 + c2j

(14)

For the given i, Eq. (13) has N unknowns with N equations. So, the weighted coef(n) ficients βik can be determined through solving this equation system. The weighted (m) coefficients β¯ik related to the y-derivatives can also be computed in a similar manner. The user-specified shape parameter ci mentioned above has a strong influence on the accuracy of numerical results. It is hard for user to adjust each ci locally to be near the optimal value. Due to the work of Shu et al.,23 this parameter

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ci can be determined adaptively through transforming the local support region to a unit circle for the two-dimensional case. This space normalization procedure results in ci = c¯Di , where Di takes the diameter of the set S(i). Doing in this way, now the user needs only to specify one c¯ instead of knot-related ci . In the present work, this constant c¯ is chosen at first simply by testing an analytical function like sin(x) + sin(y). The value c¯ of used in the present work is 4.10. 4. Problems Occurred for Treating Compressible Flows It is well-known that the Euler equations of gas dynamics are hyperbolic and hence have distinct directions of information propagation. The derivative approximation procedure described in the frame of meshless RBF-DQ method does not take the hyperbolic property of the Euler equations into account and therefore the numerical scheme based on this procedure may be unstable. Although the DQ-related methods have achieved a great success in the application to solve incompressible viscous flows (Shu et al.),23,24 they may not be suitable for simulating compressible flows involving discontinuities like shocks. One of the ways to overcome this problem is to introduce upwind mechanism in the numerical procedure of approximating spatial derivatives. In the meshless research area, the virtual control point between each pair of given knots to enforce upwind has been adopted by some researchers, examples being the work of Morinishi,13 Löhner et al.26 Sridar et al.,10 and Shu et al.27 Among various approaches, the way of a virtual mid-point between a pair of given knots is naturally considered for the sake of simplicity. According to Eq. (6), the first-order derivative of function f (x, y) with respect x to at point i can be expressed as: N ∂f βik fik + βi fi , (15) = ∂x i

k=1,k=i

where fik represents the value of the function at a mid-point (xik , yik ) between the reference knot i and the supporting knot k. With Eq. (15), Shu et al.27 applied the local upwind RBF-DQ method to simulate the one-dimensional shock tube problem and the two-dimensional shock reflection problem. The obtained numerical results agree well with exact solutions. However, it was found in Ref. 27 that the accuracy of the results is a little affected by the knot distribution. The reason is probably due to the fact that the coefficient βi exists in general. Actually, from Eq. (15), the weighting coefficients should satisfy N

βik + βi = 0 .

(16)

k=1,k=i

It means that the derivative of f (x, y) with respect to x at point i depends on the values of function at both virtual mid-points and point i itself. We can introduce artificial dissipation in the flux calculation at the mid-point by available upwind methods like using well-known Roe’s approximate Riemann solver, but the problem

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is how to enforce the upwind at point i itself. This is a key problem. In the following, we will propose some ways to bypass this problem by removing the dependence of the point i itself in the derivative approximation. 5. Two Treatments Coping with the Problem Mentioned Above 5.1. Redistribution of weighting coefficients By introducing artificial weights wik , which satisfy N

wik = 1 ,

(17)

k=1,k=i

the functional value at the reference knot (xi , yi ) can be approximated by N

f (xi , yi ) ≈

wik fik .

(18)

k=1,k=i

Substituting Eq. (18) into Eq. (15) leads to N ∂f aik fik , = ∂x

(19)

k=1,k=i

i

where aik = βik + wik βi . Through this redistribution of weighted coefficients, the derivative approximation only depends on the information at mid-points. Then the mid-point upwind strategy can be directly applied. Clearly, the new weighting coefficient aik satisfy N

aik = 0 .

(20)

k=1,k=i

Similarly, the derivative approximation in the y direction can be expressed as: N ∂f = bik fik (21) ∂y i

k=1,k=i

and the coefficient bik also satisfy bik = β¯ik + wik β¯i ,

N

bik = 0 .

(22)

k=1,k=i

The weights wik can be determined, based on the relative distance between the knots i and k. Here, we refer to the work of Ref. 28 for a list of weights. If we let wik = 1/(N − 1), Eq. (18) is actually an arithmetic average, which is tested in the present work for sake of convenience. Yet the accuracy of derivative approximation would be affected through the redistribution approach if |βi | has a big value, but numerical tests show that the magnitude of the coefficients βi is usually small enough for the points uniformly distributed in the computational domain.

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5.2. Least square local RBF-DQ method The least square local RBF-DQ method presented in this section can completely remove the contribution of flux at the local reference knot (xi , yi ). According to Eq. (12), the function f (x, y) in the supporting region of S(i) can be locally approximated with weight coefficient λij by: f (x, y) =

M

λij gij (x, y) ,

(23)

j=1

where gij (x, y) =

(x − xij )2 + (y − yij )2 + c2ij − (x − xi )2 + (y − yi )2 + c2i at

j = i, and gii (x, y) = 1 at j = i. Here (xij , yij ) represents the mid-point coordinate between the knots i and j. It should be noted that in Eq. (23) the summation number M is less than N , which means that we only select technically M points from the set S(i) to form the function approximation. In the present work, M = N −1 is tested. Considering the derivative in the x direction, from Eq. (23), we have M M ∂gij ∂gij ∂f λij λij (24) = = . ∂x ∂x ∂x j=1

i

i

j=1,j=i

i

If we can let coefficient λij depend on scattering values at all points of the set S(i) except the reference point i itself, then Eq. (24) can result in the form of Eq. (19), which does not involve the reference point i. This can be realized by a weighted least-square procedure through the following minimization problem ¯ , J(λ) ≤ J(λ)

Find λ = {λij } ∈ R , ¯ = with J(λ)

N k=1,k=i

 wik 

M

¯ = {λ ¯ ij } ∈ R , ∀λ 2

¯ij gij (xik , yik ) − fik  , λ

(25)

(26)

j=1

¯ is actually a measure of weighted error of approximating function (23) where J(λ) at all mid-points related to the set S(i). The weights wik |rik ≤¯ri = 1 and wik |rik >¯ri = (ri /rik )2 are adopted in the present work, where rik is the relative distance between the knot i and mid-point (xik , yik ), and ri is a reference distance assigned for the knot i, which could be chosen as the smallest value of all rik . In Eq. (26), the summation of the error of approximation function (23) at center point i is excluded technically, which can ensure that the resulting derivative approximations have the form of Eqs. (19) and (21). It is clear that M unknowns of the minimization problem (25) can be determined by letting ∂J/∂λij = 0. The above process can be interpreted as follows. Considering a cell enclosing the point i, the first-order derivative of a function at the cell center (with cell average like the way used in the finite volume method) can be approximated with weighted values at the cell faces. The weighting coefficients are determined by the well-known divergence theory reflecting physical conservative law. The resulting derivative approximation is also independent of center point itself.

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6. Upwind Enforced Semi-Discrete Form of Euler Equations By applying Eqs. (19) and (21), Euler equation (1) can be rewritten in a semidiscrete form as: ∂Wi + ∂t

N

aik Eik +

k=1,k=i

N

bik Fik = 0 .

(27)

k=1,k=i

Let U = au + bν and G = (ρU, ρuU + ap, ρνU + bp, (e + p)U )T , Eq. (27) can be further expressed as: ∂Wi + ∂t

N

Gik = 0 .

(28)

k=1,k=1

Using the well-known Roe’s Flux function,29 Gik can be written as: Gik =

1 + − + − [G(Wik ) + G(Wik ) − |A|(Wik − Wik )] , 2

(29)

where Jacobian matrix A = ∂G/∂W may be diagonalized as A = TΛT−1 , where the columns of T are the eigenvectors of A, and Λ is diagonal matrix of the eigenvalues. The notation of |A| = T|Λ|T−1 means the matrix containing the absolute values of the eigenvalues. In order to avoid occurring of nonlinear instabilities near sonic zones, the minimal absolute values of the eigenvalues are limited according to the work of Swanson et al.30 ˜ = (ρ, u, ν, p)T and P = ∂W/∂ W, ˜ then the difference (W+ − W− ) can Let W ik ik be determined by: + − ˜ + −W ˜ −), − Wik = P(W Wik ik ik

(30)

˜ − are computed with flux limiters φ− and φ+ as follows: ˜ + and W where W ik ik ˜+ =W ˜ i + 1 φ− ∇W ˜ i• r ik , W ik 2

(31)

˜− =W ˜ k − 1 φ+ ∇W ˜ k• r ik . W ik 2

(32)

˜ is directly Here r ik is the vector pointing from knot i to k. The gradient ∇W ˜ ˜ ˜ computed according to Eqs. (19) and (21). Let ∆W = Wk − Wi , the flux limiter φ− used in this paper can be expressed as29

φ− =

˜ + |∇W ˜ i • r ik ∆W| + ε) ˜ i • r ik ∆W (∇W . ˜ 2 + (∇W ˜ i• r ik )2 + ε] [(∆W)

(33)

A small value of ε = 10−12 is used to prevent division by zero in the smooth regions where the differences are very small. The φ+ can be formulated in a similar way (see Ref. 29 for details). The elements in the Jacobian matrix A are calculated by using the well-known Roe’s averages.29

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Through the above process, the artificial dissipation can be effectively introduced in the flux at the mid-pint. The semi-discrete equation (28) can be solved by explicit schemes such as Runge–Kutta method. But the convergence rate of explicit schemes is very slow. To speed up the convergence, in the following, we will dedicate to develop an implicit solver through the implementation of LU-SGS algorithm in the frame of meshless RBF-DQ method described above. 7. Implementation of Implicit Matrix-Free LU-SGS Method Referring to the work of Jameson and Yoon,31 the flux function, Gik may be linearized by setting − ≈ Gnik + A+ Gn+1 ik ik (Wi )δWi + Aik (Wk )δWk ,

(34)

where n is the time level, δ( ) = ( )n+1 − ( )n is the correction. Matrices A± ik are constructed by setting31 A± ik =

1 (Aik ± λik I) , 2

(35)

where λik is the eigenvalue of Jacobian matrix A, and I is the identity matrix. Then, the time backward, implicit scheme of Euler equation can be expressed by:   N N 1 n   I+ A+ A− (36) ik (Wi ) δWi + ik (Wk )δWk = R(Wi ) , ∆ti k=1,k=i

k=1,k=i

where ∆ti is local time step and R is the residual. Due to the properties of the coefficients aik and bik (see Eqs. (20) and (22)), the Jacobian matrix Aik should also satisfy N

Aik (Wi ) = 0 .

(37)

k=1,k=i

Thus, Eq. (36) can be rewritten as:   N 1 1  + λik (Wi ) IδWi + ∆ti 2 k=1,k=i

N k=1,k=i

n A− ik (Wk )δWk = R(Wi ) ,

(38)

which can be easily solved by a LU-SGS method32 with the following forward and backward sweeps   ∗ R(Win ) − A− δWi∗ = D−1 , (39a) i ik (Wk )δWk k∈L(i)

δWi = δWi∗ − D−1 i

k∈U(i)

∗ A− ik (Wk )δWk ,

(39b)

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where L(i) and U (i) are the lower and upper subsets of the set S(i), respectively. Di is denoted by  1 1 + Di =  ∆ti 2

N

 λik (Wi ) I .

(40)

k=1,k=i

In the forward and backward sweeps of Eq. (39), the computation involves the product of Jacobian matrix and correction vector, which may be approximated by the increment of flux vector due to the work of Sharov and Nakahashi33 by setting AδW ≈ δG. Thus the present LU-SGS method is actually a matrix-free approach.

(a)

(b) Fig. 2. Knot distributions around a NACA 0012 airfoil used in present computation. (a) Knots from a C-type mesh and (b) Knots from adaptive Cartesian cells.

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Reference Point Supporting Points

5-Points Fig. 3.

7-Points

9-Points

Different supporting sets tested in present studies.

8. Numerical Results and Discussion The transonic flow over a NACA 0012 airfoil is chosen to validate the proposed least square local RBF-DQ method. Unlike the conventional mesh-based methods, in this work, we only need to distribute points in the computational domain. This feature gives us a great flexibility to put the points near the boundary in such a way that the boundary conditions can be implemented conveniently. One direct choice is to put points near the boundaries like the grid nodes introduced in the structured mesh. Then the boundary conditions can be directly implemented according to any familiar one used in the structured mesh. This idea is adopted in the present work. The implementation of boundary conditions on the solid boundary and far field non reflecting boundary has been well studied,34 and is adopted in the present work. As mentioned above, mesh-free method is open to use any existing means to generate knots in the computational domain. In the present paper, two types of mesh knots, namely c-type mesh knots (8669 points) and adaptive Cartesian knots (7762 points) are used as shown in Fig. 2. In order to evaluate the performance of two treatments described in Sec. 5, three test cases with different supporting sets, namely 5-points, 7-points and 9-points, are selected from c-type mesh knots as shown in Fig. 3, where the case of 7-points is a sever case due to very poor centersymmetry. Figures 4 and 5 show the numerical results of the three test cases. The convergence histories presented in Fig. 4 show that computations with different supporting sets do affect the behavior of convergence histories, but the extent of affection is quite different with different treatments. The least square local RBFDQ approach is much better than the redistribution of weighted coefficients. We can see that the convergences of the former are quite similar with different supporting sets, even with the set of sever 7-points, but the later is notably different in the history of convergence and even has convergence difficulty for the test case of 7-points. This phenomenon is also readable from the corresponding results of surface pressure coefficients as shown in Fig. 5. Thus the least square local RBFDQ approach is recommended for simulation of compressible flows with shocks. In order to get further impression with this approach, typical Mach contours for the

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5-points 7-points 9-points

5-points 7-points 9-points

(a)

451

(b)

Fig. 4. Comparisons of convergence histories of the two approaches. (a) Redistribution of weighting coefficients and (b) Least-square local RBF-DQ.

(a)

(b)

Fig. 5. Comparisons of pressure coefficients of the two approaches. (a) Redistribution of weighting coefficients and (b) Least-square local RBF-DQ.

cases of Mach number 0.8 with zero angle of attack and Mach number 0.85 with 1.0 degree of angle of attack are presented in Fig. 6. The contours are computed using the knots from unstructured adaptive Cartesian cells. Clearly, the strong and weak shock waves are well captured. Figure 7 displays the numerical results of surface pressure coefficient around the NACA0012 airfoil for both subsonic and transonic flow cases. The numerical results of Jameson34 are also included in Fig. 7

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Fig. 6.

Fig. 7.

Typical Mach contours.

Comparison of pressure coefficients around the surface of NACA0012 airfoil.

for comparison. Obviously, the present mesh-free results agree very well with those of Jameson.34 Numerical experiments show that the present LU-SGS implicit method is over 150 times faster than its explicit counterpart (four-step Runge–Kutta scheme without any acceleration measures like residual smoothing) in view of CPU time consuming for the particular test cases mentioned above. Typical convergences between implicit and explicit solvers are presented in Fig. 8, which reveals that the implicit scheme is much better to decrease the residual and the convergence of the explicit method deteriorates dramatically in the later iterations. From the obtained numerical results, it seems that the physical conservation laws are satisfied by the method since both the shock position and strength are well captured. However, it is very difficult to prove it in mathematics since the knots can be randomly distributed and there are so many cases for knot distribution. Hopefully, this problem can be resolved in the future by mathematicians.

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Explicit (Runge–Kutta)

Fig. 8.

Typical convergence histories of implicit and explicit schemes.

9. Conclusions In this paper, an implicit LU-SGS scheme is developed in the frame of local RBFDQ method for solving two-dimensional compressible Euler equations. The resulting method has the mesh-free and matrix-free features. Two approaches, i.e., the redistribution of weighting coefficients and the least square local RBF-DQ approach, are proposed to eliminate the contribution from the reference point itself in the derivative approximation. Numerical experiments show that both approaches can capture main features of transonic flows like shock position and strength properly, but the weighted least square local RBF-DQ approach behaves much better in the sense of more flexibility in the selection of local supporting points and less sensitive to the shape parameter of RBFs. Acknowledgments This research is partially supported by Natural Science Foundation of China (10372043) and Chinese Aeronautical Foundation for Scientific Research (02A52002). References 1. Z. J. Wang, Computer and Fluids 27, 529 (1998). 2. A. Jameson, Progress in Aerospace Sciences 37, 197 (2001). 3. L. B. Lucy, The Astro. J. 8, 1013 (1977).

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4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

00732

H. Q. Chen & C. Shu

B. Nayroles, G. Touzot and P. Villon, Comput. Mech. 10, 307 (1992). T. Belytschko, Y. Y. Lu and L. Gu, Int. J. Numer. Meth. Eng. 37, 229 (1994). W. Liu, S. Jun and Y. Zhang, Int. J. Numer. Meth. Fluids 20, 1081 (1995). C. A. Duarte and J. T. Odenm, Hp clouds-a meshless method to solve boundary-value problems, TICAM Report 95-05. E. Onate, S. Idelsohn, O. C. Zienkiewicz and R. L. Taylor, Int. J. Numer. Meth. Eng. 39, 3839 (1996). K. C. Chung, Numer. Heat Transfer 4, 345 (1981). D. Sridar and N. Balakrishnan, J. Comput. Phys. 189, 1 (2003). T. J. Batina, A gridless Euler/Navier–Stokes solution algorithm for complex aircraft applications, AIAA Paper No. 93-0333 (1993). K. Morinishi, Effective accuracy and conservation consistency of gridless type solver, Computational Fluid Dynamics 2000, ed. N. Satofuka (Springer, Berlin, 2000). K. Morinishi, CFD J. (Proc. 8th ISCFD 1999 at ZARM Bremen) (2000), pp. 551–560. E. J. Kansa, Comput. Math. Appl. 19, 127 (1990). E. J. Kansa, Comput. Math. Appl. 19, 147 (1990). B. Forberg and T. A. Driscoll, Comput. Math. Appl. 43, 413 (2002). Y. C. Hon and Z. M. Wu, Int. J. Numer. Meth. Eng. 48, 1187 (2000). C. S. Chen, C. A. Brebbia and H. Power, Boundary Elements 20, 495 (1998). G. E. Fasshauer, Solving partial differential equations by collocation with radial basis functions, Surface Fitting and Multiresolution Methods, eds. A. L. Mehaute, C. Rabut and L. L. Schumaker (Vanderbilt University Press, 1997) p. 131. W. Chen, Engng. Anal. Bound. Elem. 26, 577 (2002). W. Chen and M. Tanaka, Int. J. Nonlinear Sci. Numer. Simul. 1, 145 (2000). W. Chen and M. Tanaka, Comput. Math. Appl. 43, 379 (2002). C. Shu, H. Ding and K. S. Yeo, Comput. Meth. Appl. Mech. Eng. 192, 941 (2003). C. Shu and B. E. Richards, Int. J. Numer. Meth. Fluids 15, 791 (1992). J. Yoon, Siam J. Math. Anal. 33, 946 (2001). R. L¨ ohner, C. Sacco, E. O˜ nate and S. Idelsohn, Int. J. Numer. Meth. Engrg. 53, 1765 (2002). C. Shu, H. Ding, H. Q. Chen and T. G. Wang, Comput. Meth. Appl. Mech. Eng., in press. H. Ding, C. Shu, K. S. Yeo and D. Xu, Comput. and Fluids 33, 137 (2004). P. L. Roe, J. Comput. Phys. 23, 357 (1981). R. C. Swanson and E. Turkel, J. Comput. Phys. 101, 292 (1992). A. Jameson and S. Yoon, AIAA J. 25, 929 (1987). A. Jameson and E. Turkel, Math. Comput. 37, 385 (1981). D. Sharov and K. Nakahashi, Reordering of 3-D hybrid unstructured grids for vectorized LU-SGS Navier–Stokes computations, AIAA Paper 97-2102 (1997). A. Jameson, Appl. Math. Comput. 13, 327 (1983).