- Email: [email protected]
A moving least squares meshless method for solving the generalized Kuramoto-Sivashinsky equation
Alexandria Engineering Journal (2016) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
A moving least squares meshless method for solving the generalized Kuramoto-Sivashinsky equation E. Dabboura, H. Sadat *, C. Prax Institut PPRIME, CNRS, Universite´ de Poitiers, ENSMA UPR 3346, De´partement Fluides, Thermique, Combustion, ESIP, Campus Sud 40, avenue du recteur Pineau, 86022 Poitiers Cedex, France Received 27 January 2015; revised 18 July 2016; accepted 24 July 2016
KEYWORDS Meshless; Kuramoto; Sivashinsky; Least squares
Abstract We use a moving least squares meshless method to solve the nonlinear KuramotoSivashinsky equation. The accuracy of the method is demonstrated by three test problems for which the numerical results are found to be in excellent agreement with analytical solution. Ó 2016 Faculty of Engineering, Alexandria University Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction The Kuramoto-Sivashinsky (KS) [1,2] equation is a nonlinear fourth order partial differential equation that has been proposed in the seventies for describing turbulence in reactive systems and diffusive instabilities in laminar flame fronts. Depending on the associated parameters this equation can be seen as an example of complex spatiotemporal dynamics leading to chaotic behavior. It has therefore been the subject of extensive analytical and numerical studies. Finite difference, finite volume and finite element methods have been used for the spatial discretization [3–7]. In order to use a Lattice Boltzmann method Lai and Ma [8] have proposed to construct a five velocity lattice Boltzmann model by introducing an amending function. Their results were found to be very accurate. A meshfree method using radial basis function (RBF) for the space discretization has also been proposed recently [9]. In this work, we introduce the moving least squares meshless method to solve the KS equation. This approach * Corresponding author. E-mail address: [email protected] (H. Sadat). Peer review under responsibility of Faculty of Engineering, Alexandria University.
has already been used to solve several problems in heat transfer and related fluid flow problems ranging from natural and forced convection to radiative transfer in participating media [10–17]. In all these works second order in space meshless discretization has been used successfully. In [18] we have considered higher order meshless approximations to solve second order diffusion and transport-diffusion type equations. This meshless discretization technique can be found in the literature under multiple denominations as in [19,20]. Under several other formulations, the meshless approaches are still the subject of numerous developments [21–26]. In the following sections, the fourth order meshless approximation method is first described. The same three numerical cases studied by Lai and Ma [8] are then considered. It is found that the meshless results are in excellent agreement with the exact solutions. 2. Fourth order in space meshless method Let us consider the following Taylor development at order 4 around a point M of coordinate x for a neighbor point Mi of coordinate xi of the space discretization:
http://dx.doi.org/10.1016/j.aej.2016.07.024 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: E. Dabboura et al., A moving least squares meshless method for solving the generalized Kuramoto-Sivashinsky equation, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.024
2
E. Dabboura et al. du ðxi xÞ2 d2 u ðxi xÞ3 d3 u þ þ dx 2! 3! dx2 dx3 ðxi xÞ4 d4 u þ þ OðDx4 Þ 4! dx4
ui ðxi Þ ¼ u þ ðxi xÞ
ð1Þ
This can be written in more compact form as follows: ui ðxi Þ ¼ hpðMi ; MÞi haM iT
ð2Þ
hpðMi ; MÞi is the line vector of the generalized polynomial basis and haM iT the transpose vector of the generalized variables of the approximation which are the successive derivatives. Hence we have the following: 2
3
4
hpðMi ; MÞi ¼ h1; ðxi xÞ; ðxi xÞ ; ðxi xÞ ; ðxi xÞ i
ð3Þ
and the successive derivatives of the unknown function at point M are expressed in terms of the different values ui of the function in the chosen neighboring nodes. 1 If haji represents the jth line of the inverse matrix AM , the derivatives now simply write the following: X @u xðMi ; MÞha1 ihpðMi ; MÞiT :ui ð11Þ ¼ @x M Mi 2V
X @2u ¼ 2! xðMi ; MÞha2 ihpðMi ; MÞiT :ui @x2 M 2VM
ð12Þ
X @3u ¼ 3! xðMi ; MÞha3 ihpðMi ; MÞiT :ui 3 @x M 2VM
ð13Þ
X @4u ¼ 4! xðMi ; MÞha4 ihpðMi ; MÞiT :ui 4 @x M M 2V
ð14Þ
i
i
and haM iT ¼ ha0 ; a1 ; a2 ; a3 ; a4 iT
ð4Þ
ui
If the discrete values of function u are supposed to be known on n neighboring nodes Mi, one can relate the approximations of the successive derivatives to the discrete values ui , by minimizing the following quadrature error: IðaM Þ ¼
N n X
2 o xðMi ; MÞ ui hpðMi ; MÞihaM iT
ð5Þ
i¼1
where x is a positive weight function of compact support centered at the considered point M and rapidly decaying. The support of this function will define the number of neighboring nodes used for the approximation. One can now minimize the quadratic form by writing the following: @IðaM Þ ¼ 0 for i ¼ 0; . . . 4 ð6Þ @ai This leads to the system: 8 9 u > > > > > > > du > > > > dx > > > > > M < d2 u = 2 ¼ hBM iT A 2!dx > > > d3 u > > > > > > 3!dx3 > > > > > d4 u > > : ;
ð7Þ
4!dx4
where [A ] and hBMiT are defined by the following: hBM iT M
½AM ¼
N X xðMi ; MÞhpðMi ; MÞiT hpðMi ; MÞi
ð8Þ
i¼1 N X hBM iT ¼ xðMi ; MÞhpðMi ; MÞiT :ui
i
The present collocation meshless method uses the strong formulation of the equation to solve in the sense that at each calculation point, all the derivatives appearing in the equation are replaced by their approximations given by previous expressions (11)–(14) leading thus to an algebraic equation at the point and finally to a system of N algebraic equations if N calculation points are used in the spatial discretization. Boundary conditions are introduced to the algebraic system which is then solved once in a steady state problem or at each time step in an unsteady problem which is the case herein. For the time discretization we used a simple Euler implicit scheme although more accurate schemes could be used. The weighting function can have several forms (triangular, Hanning, exponential. . .). In this work the following Gaussian function has been employed: r 2 xðrÞ ¼ Exp ln ðeÞ S where r = |MMi| represents the distance between points M and his neighbors Mi and where S is the size of the function support. It is important to note that the weight function must be sufficiently large to enclose a number of nodes at least equal to the number of generalized variables. Finally, our previous works [10–17] have shown that e value can be chosen in the range [103–109], depending on the number of selected nodes. In this work, a constant value of 106 is chosen. 3. Application to the Kuramoto-Sivashinsky equation
ð9Þ
i¼1
If matrix [AM] is not singular, the system (7) can be inverted: 8 9 u > > > > > > du > > > > > dx > > > ( ) > n < d2 u > = X M 1 T 2 ¼ A xðMi ; MÞ hpðMi ; MÞi ui 2!dx > > > i¼1 d3 u > > > > > 3 > 3!dx > > > > > d4 u > > : ; 4!dx4
ð10Þ
We present in this section the results obtained by the previous meshless method when applied to the following KS equation: @u @u @2u @3u @4u þu þa 2þb 3þc 4 ¼0 @t @x @x @x @x
ð15Þ
where a, b and c are problem dependant constants. The three particular problems studied in [8] are used to test the present approach. As in [8] the global relative error (GRE) is introduced for testing the precision: P ju ðxi ; tÞ uðxi ; tÞj GRE ¼ i P ð16Þ i juðxi ; tÞj where u* is the numerical solution and u the exact solution.
Please cite this article in press as: E. Dabboura et al., A moving least squares meshless method for solving the generalized Kuramoto-Sivashinsky equation, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.024
A moving least squares meshless method
3
3.1. First example We have first considered the case where (a = 1, b = 1, c = 1). The equation and its analytical solution are as follows: @u @u @ 2 u @ 4 u þu þ þ ¼0 @t @x @x2 @x4 rffiffiffiffiffi 15 11 u ðx; tÞ ¼ b þ ½9 tanhðkðx bt xo ÞÞ 19 19 þ11tanh3 ðkðx bt xo ÞÞ
ð17Þ
ð18Þ
We present the results obtained for the following parameqffiffiffiffi and xo = 12. ters: b = 5, k ¼ 12 11 19 The studied domain [30, 30] has been discretized with 601 points so that Dx = 0.1. The time step has been set to Dt = 0.0001. At the two limits of the studied domain (x = 30 and x = 30), two boundary conditions (Dirichlet and Neumann) have been used. They are both deduced from the analytical exact solution. The Dirichlet conditions are imposed to the boundary points while an algebraic expression is deduced from the Neumann conditions at each side and applied to the first interior point. The exact analytical solution and the meshless results are shown in Fig. 1 at times equal to 1, 2, 3, and 4 s. It can be seen that the meshless results are perfectly superposed to the exact solutions. The global relative error (GRE) is reported in Table 1 together with the results obtained by the Lattice Boltzmann method in [8]. One can see that the meshless method is more accurate than the LB method for this particular problem. The good quality of the numerical solution is maintained at time 6 s and 8 s where the GRE is 5.4e4 and 1.9e3.
Table 1
Global relative error for test 1.
T
1
2
3
4
GRE [8] GRE [this work]
67923 104 44905 105
11503 103 75401 105
15941 103 11464 104
20075 103 15882 104
15 pffiffiffiffiffi ½3 tanhðkðx bt xo ÞÞ 19 19 þtanh3 ðkðx bt xo ÞÞ
u ðx; tÞ ¼ b þ
ð20Þ
ffi The following parameters have been used: b = 5, k ¼ 2p1ffiffiffi 19 and xo = 25. 1000 nodes have been distributed on the studied domain [50, 50] leading to a space increment of Dx = 0.1. Once again a time step Dt = 0.0001 has been chosen. The meshless and exact solutions are presented in Fig. 2 for t = 6, t = 8, t = 10 and t = 12 s. Once again, a very good agreement is obtained. The global relative error (GRE) at different times is reported in Table 2. Once again, the meshless method seems to be more accurate than the LB method. 3.3. Third example In this final example, the third order derivative is introduced. The equation and its solution write the following: @u @u @ 2 u @3u @4u þu þ 2þ4 3þ 4 ¼0 @t @x @x @x @x
ð21Þ
u ðx; tÞ ¼ b þ 9 15½tanhðkðx bt xo ÞÞ
þtanh2 ðkðx bt xo ÞÞ tanh3 ðkðx bt xo ÞÞ ð22Þ
3.2. Second example In this second example, the equation to be solved and its analytical solution are as follows: @u @u @ 2 u @ 4 u þu þ ¼0 @t @x @x2 @x4
5.4
ð19Þ
5.3
7
5.2
6.5
t=6 t=8
U
5.5
t=1
5.1
t=10
5
t=12
t=2
U
6
5 4.9
4.5 4.8
4 t=3 3.5 3 -30
4.7
t=4 -20
-10
0
10
20
30
X
Figure 1 Numerical and analytical solutions at different times. The solid lines represent the analytical solutions.
4.6 -50
0
50
X
Figure 2 Numerical and analytical solutions at different times. The solid lines represent the analytical solutions.
Please cite this article in press as: E. Dabboura et al., A moving least squares meshless method for solving the generalized Kuramoto-Sivashinsky equation, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.024
4
E. Dabboura et al. meshless method which deserves much attention than it actually receives.
Table 2
Global relative error for test 2.
T
6
8
10
12
GRE [8] GRE [this work]
78808 106 16669 106
95324 106 20397 106
10891 105 23718 106
11793 105 26287 106
18
U
16 14
t=1
12
t=2
10
t=3
8
t=4
6 4 2 0 -2 -30
-20
-10
0
10
20
30
X
Figure 3 Numerical and analytical solutions at different times. The solid lines represent the analytical solutions.
Table 3
Global relative error for test 3.
t
1
2
3
4
GRE [8] GRE [this work]
25945 102 04975 102
27959 102 07052 102
26701 102 16426 102
35172 102 29299 102
The different parameters are as follows: b = 6, k = 0.5, xo = 10, Dx = 0.1, Dt = 0.0001. The studied domain [30, 30] is discretized with 600 nodes. The meshless and exact solutions are presented in Fig. 3 for different times and the GRE is reported in Table 3. Once again, the agreement is very good and the method seems to be as accurate or even more accurate than the LB method. 4. Conclusion We have proposed a numerical treatment of the KuramotoSivashinsky equation by using a moving least squares approximation based meshless method. Three examples have been considered. The numerical results that are presented graphically show that the method is very accurate. The global relative error has been calculated and compared to that given in the open literature. This is another field of application of this
References [1] G.I. Sivashinsky, Non linear analysis of hydrodynamic instability in laminar flames. Part I: Derivation of the basic equations, Acta Astronaut. 4 (1977) 1176–1206. [2] Y. Kuramato, Diffusion induced chaos in reactions systems, Prog. Theor. Phys. Suppl. 64 (1978) 346–367. [3] G.D. Akrivis, Finite difference discretization of the Kuramoto– Sivashinsky equation, Numerische Mathematik 63 (1) (1992) 1– 11. [4] P. Brunet, Stabilized Kuramoto–Sivashinsky equation: a useful model for secondary instabilities and related dynamics of experimental one-dimensional cellular flows, Phys. Rev. E 76 (2007) 017204. [5] S.M. Choo, Finite element schemes for generalized Cahn– Hilliard and Kuramoto–Sivashinsky equations, J. Appl. Math. Comput. 21 (2006) 531–538. [6] G. Akrivis, High order finite element methods for the Kuramoto–Sivashinsky equation, RAIRO, Mode´lisation mathe´matique et analyse nume´rique, tome 30 (1996) 157–183. [7] L. Cueto-Felgueroso, J. Peraire, A time-adaptive finite volume method for the Cahn–Hilliard and Kuramoto–Sivashinsky equations, J. Comput. Phys. 227 (24) (2008) 9985–10017. [8] Huilin Lai, Changfeng Ma, Lattice Boltzmann method for the generalized Kumamoto–Sivashinsky equation, Physica A 388 (2009) 1405–1412. [9] U. Marjan, H. Sirajul, I. Siraj, A mesh-free numerical method for solution of the family of Kuramoto–Sivashinsky equations, Appl. Math. Comput. 212 (2) (2009) 458–469. [10] C. Prax, H. Sadat, P. Salagnac, Diffuse approximation method for solving natural convection in porous media, Transp. Porous Media 22 (2) (1996) 215–223. [11] C. Prax, H. Sadat, Mise en œuvre d’une me´thode d’approximation diffuse adaptative pour la re´solution des e´quations de diffusion et de transport, Int. J. Therm. Sci. 37 (1) (1998). [12] C. Prax, P. Salagnac, H. Sadat, Diffuse approximation method and control volume finite element method: a comparative study, Numer. Heat Transfer, Part B (1998). [13] H. Sadat, S. Couturier, Performance and accuracy of a meshless method for laminar natural convection, Numer. Heat Transfer, Part B (2000). [14] T. Sophy, H. Sadat, C. Prax, A meshless formulation for threedimensional laminar natural convection, Numer. Heat Transfer, Part B 41 (5) (2002) 433–445. [15] T. Sophy, H. Sadat, R. Bouard, Numerical study of the flow over a half-circular cylinder by a diffuse approximation-based collocation method, C.R. Mec. 330 (3) (2002) 193–198. [16] H. Sadat, On the use of a meshless method for solving radiative transfer with the discrete ordinates formulations, J. Quant. Spectrosc. Radiat. Transfer 101 (2) (2006) 263–268. [17] C.-A. Wang, H. Sadat, V. Ledez, D. Lemonnier, Meshless method for solving radiative transfer problems in complex twodimensional and three-dimensional geometries, Int. J. Therm. Sci. 49 (12) (2010) 2282–2288. [18] C. Prax, H. Sadat, E. Dabboura, Evaluation of high order versions of the diffuse approximate meshless method, Appl. Math. Comput. 186 (2) (2007) 1040–1053. [19] F. Urena, E. Salete, J.J. Benito, L. Gavete, Solving third- and fourth-order partial differential equations using GFDM: application to solve problems of plates, Int. J. Comput. Math. 89 (2012) 366–376. [20] C.M. Fan, Y.K. Huang, P.W. Li, C.L. Chiu, Application of the generalized finite-difference method to inverse biharmonic
Please cite this article in press as: E. Dabboura et al., A moving least squares meshless method for solving the generalized Kuramoto-Sivashinsky equation, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.024
A moving least squares meshless method boundary-value problems, Numer. Heat Transfer, Part B 65 (2014) 129–154. [21] M. Lakestani, M. Dehghan, Numerical solutions of the generalized Kuramoto–Sivashinsky equation using B-spline functions, Appl. Math. Model. 36 (2) (2012) 605–617. [22] M. Dehghan, R. Salehi, A meshless based numerical technique for traveling solitary wave solution of Boussinesq equation, Appl. Math. Model. 36 (2012) 1939–1956. [23] R. Salehi, M. Dehghan, A generalized moving least square reproducing kernel method, J. Comput. Appl. Math. 249 (2013) 120–132.
5 [24] R. Salehi, M. Dehghan, A moving least square reproducing polynomial meshless method, Appl. Numer. Math. 69 (2013) 34–58. [25] M. Dehghan, M. Abbaszadeh, A. Mohebbi, The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate, J. Comput. Appl. Math. 286 (2015) 211–231. [26] M. Dehghan, M. Abbaszadeh, A. Mohebbi, The numerical solution of the two-dimensional sinh-Gordon equation via three meshless methods, Eng. Anal. Boundary Elem. 51 (2015) 220– 235.
Please cite this article in press as: E. Dabboura et al., A moving least squares meshless method for solving the generalized Kuramoto-Sivashinsky equation, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.024
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
A moving least squares meshless method for solving the generalized Kuramoto-Sivashinsky equation E. Dabboura, H. Sadat *, C. Prax Institut PPRIME, CNRS, Universite´ de Poitiers, ENSMA UPR 3346, De´partement Fluides, Thermique, Combustion, ESIP, Campus Sud 40, avenue du recteur Pineau, 86022 Poitiers Cedex, France Received 27 January 2015; revised 18 July 2016; accepted 24 July 2016
KEYWORDS Meshless; Kuramoto; Sivashinsky; Least squares
Abstract We use a moving least squares meshless method to solve the nonlinear KuramotoSivashinsky equation. The accuracy of the method is demonstrated by three test problems for which the numerical results are found to be in excellent agreement with analytical solution. Ó 2016 Faculty of Engineering, Alexandria University Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction The Kuramoto-Sivashinsky (KS) [1,2] equation is a nonlinear fourth order partial differential equation that has been proposed in the seventies for describing turbulence in reactive systems and diffusive instabilities in laminar flame fronts. Depending on the associated parameters this equation can be seen as an example of complex spatiotemporal dynamics leading to chaotic behavior. It has therefore been the subject of extensive analytical and numerical studies. Finite difference, finite volume and finite element methods have been used for the spatial discretization [3–7]. In order to use a Lattice Boltzmann method Lai and Ma [8] have proposed to construct a five velocity lattice Boltzmann model by introducing an amending function. Their results were found to be very accurate. A meshfree method using radial basis function (RBF) for the space discretization has also been proposed recently [9]. In this work, we introduce the moving least squares meshless method to solve the KS equation. This approach * Corresponding author. E-mail address: [email protected] (H. Sadat). Peer review under responsibility of Faculty of Engineering, Alexandria University.
has already been used to solve several problems in heat transfer and related fluid flow problems ranging from natural and forced convection to radiative transfer in participating media [10–17]. In all these works second order in space meshless discretization has been used successfully. In [18] we have considered higher order meshless approximations to solve second order diffusion and transport-diffusion type equations. This meshless discretization technique can be found in the literature under multiple denominations as in [19,20]. Under several other formulations, the meshless approaches are still the subject of numerous developments [21–26]. In the following sections, the fourth order meshless approximation method is first described. The same three numerical cases studied by Lai and Ma [8] are then considered. It is found that the meshless results are in excellent agreement with the exact solutions. 2. Fourth order in space meshless method Let us consider the following Taylor development at order 4 around a point M of coordinate x for a neighbor point Mi of coordinate xi of the space discretization:
http://dx.doi.org/10.1016/j.aej.2016.07.024 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: E. Dabboura et al., A moving least squares meshless method for solving the generalized Kuramoto-Sivashinsky equation, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.024
2
E. Dabboura et al. du ðxi xÞ2 d2 u ðxi xÞ3 d3 u þ þ dx 2! 3! dx2 dx3 ðxi xÞ4 d4 u þ þ OðDx4 Þ 4! dx4
ui ðxi Þ ¼ u þ ðxi xÞ
ð1Þ
This can be written in more compact form as follows: ui ðxi Þ ¼ hpðMi ; MÞi haM iT
ð2Þ
hpðMi ; MÞi is the line vector of the generalized polynomial basis and haM iT the transpose vector of the generalized variables of the approximation which are the successive derivatives. Hence we have the following: 2
3
4
hpðMi ; MÞi ¼ h1; ðxi xÞ; ðxi xÞ ; ðxi xÞ ; ðxi xÞ i
ð3Þ
and the successive derivatives of the unknown function at point M are expressed in terms of the different values ui of the function in the chosen neighboring nodes. 1 If haji represents the jth line of the inverse matrix AM , the derivatives now simply write the following: X @u xðMi ; MÞha1 ihpðMi ; MÞiT :ui ð11Þ ¼ @x M Mi 2V
X @2u ¼ 2! xðMi ; MÞha2 ihpðMi ; MÞiT :ui @x2 M 2VM
ð12Þ
X @3u ¼ 3! xðMi ; MÞha3 ihpðMi ; MÞiT :ui 3 @x M 2VM
ð13Þ
X @4u ¼ 4! xðMi ; MÞha4 ihpðMi ; MÞiT :ui 4 @x M M 2V
ð14Þ
i
i
and haM iT ¼ ha0 ; a1 ; a2 ; a3 ; a4 iT
ð4Þ
ui
If the discrete values of function u are supposed to be known on n neighboring nodes Mi, one can relate the approximations of the successive derivatives to the discrete values ui , by minimizing the following quadrature error: IðaM Þ ¼
N n X
2 o xðMi ; MÞ ui hpðMi ; MÞihaM iT
ð5Þ
i¼1
where x is a positive weight function of compact support centered at the considered point M and rapidly decaying. The support of this function will define the number of neighboring nodes used for the approximation. One can now minimize the quadratic form by writing the following: @IðaM Þ ¼ 0 for i ¼ 0; . . . 4 ð6Þ @ai This leads to the system: 8 9 u > > > > > > > du > > > > dx > > > > > M < d2 u = 2 ¼ hBM iT A 2!dx > > > d3 u > > > > > > 3!dx3 > > > > > d4 u > > : ;
ð7Þ
4!dx4
where [A ] and hBMiT are defined by the following: hBM iT M
½AM ¼
N X xðMi ; MÞhpðMi ; MÞiT hpðMi ; MÞi
ð8Þ
i¼1 N X hBM iT ¼ xðMi ; MÞhpðMi ; MÞiT :ui
i
The present collocation meshless method uses the strong formulation of the equation to solve in the sense that at each calculation point, all the derivatives appearing in the equation are replaced by their approximations given by previous expressions (11)–(14) leading thus to an algebraic equation at the point and finally to a system of N algebraic equations if N calculation points are used in the spatial discretization. Boundary conditions are introduced to the algebraic system which is then solved once in a steady state problem or at each time step in an unsteady problem which is the case herein. For the time discretization we used a simple Euler implicit scheme although more accurate schemes could be used. The weighting function can have several forms (triangular, Hanning, exponential. . .). In this work the following Gaussian function has been employed: r 2 xðrÞ ¼ Exp ln ðeÞ S where r = |MMi| represents the distance between points M and his neighbors Mi and where S is the size of the function support. It is important to note that the weight function must be sufficiently large to enclose a number of nodes at least equal to the number of generalized variables. Finally, our previous works [10–17] have shown that e value can be chosen in the range [103–109], depending on the number of selected nodes. In this work, a constant value of 106 is chosen. 3. Application to the Kuramoto-Sivashinsky equation
ð9Þ
i¼1
If matrix [AM] is not singular, the system (7) can be inverted: 8 9 u > > > > > > du > > > > > dx > > > ( ) > n < d2 u > = X M 1 T 2 ¼ A xðMi ; MÞ hpðMi ; MÞi ui 2!dx > > > i¼1 d3 u > > > > > 3 > 3!dx > > > > > d4 u > > : ; 4!dx4
ð10Þ
We present in this section the results obtained by the previous meshless method when applied to the following KS equation: @u @u @2u @3u @4u þu þa 2þb 3þc 4 ¼0 @t @x @x @x @x
ð15Þ
where a, b and c are problem dependant constants. The three particular problems studied in [8] are used to test the present approach. As in [8] the global relative error (GRE) is introduced for testing the precision: P ju ðxi ; tÞ uðxi ; tÞj GRE ¼ i P ð16Þ i juðxi ; tÞj where u* is the numerical solution and u the exact solution.
Please cite this article in press as: E. Dabboura et al., A moving least squares meshless method for solving the generalized Kuramoto-Sivashinsky equation, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.024
A moving least squares meshless method
3
3.1. First example We have first considered the case where (a = 1, b = 1, c = 1). The equation and its analytical solution are as follows: @u @u @ 2 u @ 4 u þu þ þ ¼0 @t @x @x2 @x4 rffiffiffiffiffi 15 11 u ðx; tÞ ¼ b þ ½9 tanhðkðx bt xo ÞÞ 19 19 þ11tanh3 ðkðx bt xo ÞÞ
ð17Þ
ð18Þ
We present the results obtained for the following parameqffiffiffiffi and xo = 12. ters: b = 5, k ¼ 12 11 19 The studied domain [30, 30] has been discretized with 601 points so that Dx = 0.1. The time step has been set to Dt = 0.0001. At the two limits of the studied domain (x = 30 and x = 30), two boundary conditions (Dirichlet and Neumann) have been used. They are both deduced from the analytical exact solution. The Dirichlet conditions are imposed to the boundary points while an algebraic expression is deduced from the Neumann conditions at each side and applied to the first interior point. The exact analytical solution and the meshless results are shown in Fig. 1 at times equal to 1, 2, 3, and 4 s. It can be seen that the meshless results are perfectly superposed to the exact solutions. The global relative error (GRE) is reported in Table 1 together with the results obtained by the Lattice Boltzmann method in [8]. One can see that the meshless method is more accurate than the LB method for this particular problem. The good quality of the numerical solution is maintained at time 6 s and 8 s where the GRE is 5.4e4 and 1.9e3.
Table 1
Global relative error for test 1.
T
1
2
3
4
GRE [8] GRE [this work]
67923 104 44905 105
11503 103 75401 105
15941 103 11464 104
20075 103 15882 104
15 pffiffiffiffiffi ½3 tanhðkðx bt xo ÞÞ 19 19 þtanh3 ðkðx bt xo ÞÞ
u ðx; tÞ ¼ b þ
ð20Þ
ffi The following parameters have been used: b = 5, k ¼ 2p1ffiffiffi 19 and xo = 25. 1000 nodes have been distributed on the studied domain [50, 50] leading to a space increment of Dx = 0.1. Once again a time step Dt = 0.0001 has been chosen. The meshless and exact solutions are presented in Fig. 2 for t = 6, t = 8, t = 10 and t = 12 s. Once again, a very good agreement is obtained. The global relative error (GRE) at different times is reported in Table 2. Once again, the meshless method seems to be more accurate than the LB method. 3.3. Third example In this final example, the third order derivative is introduced. The equation and its solution write the following: @u @u @ 2 u @3u @4u þu þ 2þ4 3þ 4 ¼0 @t @x @x @x @x
ð21Þ
u ðx; tÞ ¼ b þ 9 15½tanhðkðx bt xo ÞÞ
þtanh2 ðkðx bt xo ÞÞ tanh3 ðkðx bt xo ÞÞ ð22Þ
3.2. Second example In this second example, the equation to be solved and its analytical solution are as follows: @u @u @ 2 u @ 4 u þu þ ¼0 @t @x @x2 @x4
5.4
ð19Þ
5.3
7
5.2
6.5
t=6 t=8
U
5.5
t=1
5.1
t=10
5
t=12
t=2
U
6
5 4.9
4.5 4.8
4 t=3 3.5 3 -30
4.7
t=4 -20
-10
0
10
20
30
X
Figure 1 Numerical and analytical solutions at different times. The solid lines represent the analytical solutions.
4.6 -50
0
50
X
Figure 2 Numerical and analytical solutions at different times. The solid lines represent the analytical solutions.
Please cite this article in press as: E. Dabboura et al., A moving least squares meshless method for solving the generalized Kuramoto-Sivashinsky equation, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.024
4
E. Dabboura et al. meshless method which deserves much attention than it actually receives.
Table 2
Global relative error for test 2.
T
6
8
10
12
GRE [8] GRE [this work]
78808 106 16669 106
95324 106 20397 106
10891 105 23718 106
11793 105 26287 106
18
U
16 14
t=1
12
t=2
10
t=3
8
t=4
6 4 2 0 -2 -30
-20
-10
0
10
20
30
X
Figure 3 Numerical and analytical solutions at different times. The solid lines represent the analytical solutions.
Table 3
Global relative error for test 3.
t
1
2
3
4
GRE [8] GRE [this work]
25945 102 04975 102
27959 102 07052 102
26701 102 16426 102
35172 102 29299 102
The different parameters are as follows: b = 6, k = 0.5, xo = 10, Dx = 0.1, Dt = 0.0001. The studied domain [30, 30] is discretized with 600 nodes. The meshless and exact solutions are presented in Fig. 3 for different times and the GRE is reported in Table 3. Once again, the agreement is very good and the method seems to be as accurate or even more accurate than the LB method. 4. Conclusion We have proposed a numerical treatment of the KuramotoSivashinsky equation by using a moving least squares approximation based meshless method. Three examples have been considered. The numerical results that are presented graphically show that the method is very accurate. The global relative error has been calculated and compared to that given in the open literature. This is another field of application of this
References [1] G.I. Sivashinsky, Non linear analysis of hydrodynamic instability in laminar flames. Part I: Derivation of the basic equations, Acta Astronaut. 4 (1977) 1176–1206. [2] Y. Kuramato, Diffusion induced chaos in reactions systems, Prog. Theor. Phys. Suppl. 64 (1978) 346–367. [3] G.D. Akrivis, Finite difference discretization of the Kuramoto– Sivashinsky equation, Numerische Mathematik 63 (1) (1992) 1– 11. [4] P. Brunet, Stabilized Kuramoto–Sivashinsky equation: a useful model for secondary instabilities and related dynamics of experimental one-dimensional cellular flows, Phys. Rev. E 76 (2007) 017204. [5] S.M. Choo, Finite element schemes for generalized Cahn– Hilliard and Kuramoto–Sivashinsky equations, J. Appl. Math. Comput. 21 (2006) 531–538. [6] G. Akrivis, High order finite element methods for the Kuramoto–Sivashinsky equation, RAIRO, Mode´lisation mathe´matique et analyse nume´rique, tome 30 (1996) 157–183. [7] L. Cueto-Felgueroso, J. Peraire, A time-adaptive finite volume method for the Cahn–Hilliard and Kuramoto–Sivashinsky equations, J. Comput. Phys. 227 (24) (2008) 9985–10017. [8] Huilin Lai, Changfeng Ma, Lattice Boltzmann method for the generalized Kumamoto–Sivashinsky equation, Physica A 388 (2009) 1405–1412. [9] U. Marjan, H. Sirajul, I. Siraj, A mesh-free numerical method for solution of the family of Kuramoto–Sivashinsky equations, Appl. Math. Comput. 212 (2) (2009) 458–469. [10] C. Prax, H. Sadat, P. Salagnac, Diffuse approximation method for solving natural convection in porous media, Transp. Porous Media 22 (2) (1996) 215–223. [11] C. Prax, H. Sadat, Mise en œuvre d’une me´thode d’approximation diffuse adaptative pour la re´solution des e´quations de diffusion et de transport, Int. J. Therm. Sci. 37 (1) (1998). [12] C. Prax, P. Salagnac, H. Sadat, Diffuse approximation method and control volume finite element method: a comparative study, Numer. Heat Transfer, Part B (1998). [13] H. Sadat, S. Couturier, Performance and accuracy of a meshless method for laminar natural convection, Numer. Heat Transfer, Part B (2000). [14] T. Sophy, H. Sadat, C. Prax, A meshless formulation for threedimensional laminar natural convection, Numer. Heat Transfer, Part B 41 (5) (2002) 433–445. [15] T. Sophy, H. Sadat, R. Bouard, Numerical study of the flow over a half-circular cylinder by a diffuse approximation-based collocation method, C.R. Mec. 330 (3) (2002) 193–198. [16] H. Sadat, On the use of a meshless method for solving radiative transfer with the discrete ordinates formulations, J. Quant. Spectrosc. Radiat. Transfer 101 (2) (2006) 263–268. [17] C.-A. Wang, H. Sadat, V. Ledez, D. Lemonnier, Meshless method for solving radiative transfer problems in complex twodimensional and three-dimensional geometries, Int. J. Therm. Sci. 49 (12) (2010) 2282–2288. [18] C. Prax, H. Sadat, E. Dabboura, Evaluation of high order versions of the diffuse approximate meshless method, Appl. Math. Comput. 186 (2) (2007) 1040–1053. [19] F. Urena, E. Salete, J.J. Benito, L. Gavete, Solving third- and fourth-order partial differential equations using GFDM: application to solve problems of plates, Int. J. Comput. Math. 89 (2012) 366–376. [20] C.M. Fan, Y.K. Huang, P.W. Li, C.L. Chiu, Application of the generalized finite-difference method to inverse biharmonic
Please cite this article in press as: E. Dabboura et al., A moving least squares meshless method for solving the generalized Kuramoto-Sivashinsky equation, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.024
A moving least squares meshless method boundary-value problems, Numer. Heat Transfer, Part B 65 (2014) 129–154. [21] M. Lakestani, M. Dehghan, Numerical solutions of the generalized Kuramoto–Sivashinsky equation using B-spline functions, Appl. Math. Model. 36 (2) (2012) 605–617. [22] M. Dehghan, R. Salehi, A meshless based numerical technique for traveling solitary wave solution of Boussinesq equation, Appl. Math. Model. 36 (2012) 1939–1956. [23] R. Salehi, M. Dehghan, A generalized moving least square reproducing kernel method, J. Comput. Appl. Math. 249 (2013) 120–132.
5 [24] R. Salehi, M. Dehghan, A moving least square reproducing polynomial meshless method, Appl. Numer. Math. 69 (2013) 34–58. [25] M. Dehghan, M. Abbaszadeh, A. Mohebbi, The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate, J. Comput. Appl. Math. 286 (2015) 211–231. [26] M. Dehghan, M. Abbaszadeh, A. Mohebbi, The numerical solution of the two-dimensional sinh-Gordon equation via three meshless methods, Eng. Anal. Boundary Elem. 51 (2015) 220– 235.
Please cite this article in press as: E. Dabboura et al., A moving least squares meshless method for solving the generalized Kuramoto-Sivashinsky equation, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.07.024