A global Stokes method of approximated particular solutions for unsteady two-dimensional Navier–Stokes system of equations

References

• S. Albensoeder, H.C. Kuhlmann, and H.J. Rath, Multiplicity of steady two-dimensional flows in two-sided lid-driven cavities, Theoret. Comput. Fluid Dyn. 14 (2001), pp. 223–241. doi: 10.1007/s001620050138
   Web of Science ®Google Scholar
• N. Alleborn, K. Nandakumar, H. Raszillier, and F. Durst, Further contributions on the two-dimensional flow in a sudden expansion, J. Fluid Mech. 330 (1997), pp. 169–188. doi: 10.1017/S0022112096003382
   Web of Science ®Google Scholar
• S. Atluri, H. Liu, and Z. Han, Meshless local Petrov–Galerkin (MLPG) mixed collocation method for elasticity problems, CMES 14 (2006), pp. 141–152.
   Web of Science ®Google Scholar
• F. Auteri, N. Parolini, and L. Quartapelle, Numerical investigation on the stability of singular driven cavity flow, J. Comput. Phys. 183 (2002), pp. 1–25. doi: 10.1006/jcph.2002.7145
   Web of Science ®Google Scholar
• D. Barkley, M. Gomes, and R. Henderson, Three-dimensional instability in flow over a backward-facing step, J. Fluid Mech. 473 (2002), pp. 167–190. doi: 10.1017/S002211200200232X
   Web of Science ®Google Scholar
• F. Battaglia, S.J. Tavener, A.K. Kulkarni, and C.L. Merkle, Bifurcation of Low Reynolds Number Flows in Symmetric Channels, AIAA Journal 35 (1997), pp. 99–105. doi: 10.2514/2.68
   Web of Science ®Google Scholar
• H.M. Blackburn, D. Barkley, and S.J. Sherwin, Convective instability and transient growth in flow over a backward-facing step, J. Fluid Mech. 603 (2008), pp. 271–304. doi: 10.1017/S0022112008001109
   Web of Science ®Google Scholar
• V. Boppana and J. Gajjar, Global flow instability in a lid-driven cavity, Int. J. Numer. Method Fluids 62 (2010), pp. 827–853.
   Web of Science ®Google Scholar
• O. Botella, On a collocation B-spline method for the solution of the Navier-Stokes equations, Comput. Fluids 31 (2002), pp. 397–420. doi: 10.1016/S0045-7930(01)00058-5
   Web of Science ®Google Scholar
• C.-H. Bruneau and M. Saad, The 2D lid-driven cavity problem revisited, Comput. Fluids 35 (2006), pp. 326–348. doi: 10.1016/j.compfluid.2004.12.004
   Web of Science ®Google Scholar
• M. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge University Press, Cambridge, 2003.
   Google Scholar
• C.A. Bustamante, H. Power, and W.F. Florez, A global meshless collocation particular solution method for solving the two-dimensional Navier-Stokes system of equations, Comput. Math. Appl. 65 (2013), pp. 1939–1955. doi: 10.1016/j.camwa.2013.04.014
   Web of Science ®Google Scholar
• C.A. Bustamante, H. Power, and W.F. Florez, Schwarz alternating domain decomposition approach for the solution of two-dimensional Navier-Stokes flow problems by the method of approximate particular solutions, Numer. Methods Partial Diff Equ. 31 (2014), pp. 777–797. doi: 10.1002/num.21917
   Web of Science ®Google Scholar
• R.E. Carlson and T.A. Foley, The parameter R2 in multiquadric interpolation, Comput. Math. Appl. 21 (1991), pp. 29–42. doi: 10.1016/0898-1221(91)90123-L
   Web of Science ®Google Scholar
• C.S. Chen, C.M. Fan, and P.H. Wen, The method of approximate particular solutions for solving certain partial differential equations, Numer. Methods Partial Diff. Equ. 28 (2012), pp. 506–522. doi: 10.1002/num.20631
   Web of Science ®Google Scholar
• W. Cherdron, F. Durst, and J.H. Whitelaw, Asymmetric flows and instabilities in symmetric ducts with sudden expansions, J. Fluid Mech. 84 (1978), pp. 13–31. doi: 10.1017/S0022112078000026
   Web of Science ®Google Scholar
• P.P. Chinchapatnam, K. Djidjeli, and P.B. Nair, Radial basis function meshless method for the steady incompressible Navier-Stokes equations, Int. J. Comput. Math. 84 (2007), pp. 1509–1521. doi: 10.1080/00207160701308309
   Web of Science ®Google Scholar
• D. Drikakis, Bifurcation phenomena in incompressible sudden expansion flows, Phys. Fluids 9 (1997), pp. 76–87. doi: 10.1063/1.869174
   Web of Science ®Google Scholar
• E. Erturk, Numerical solutions of 2-D steady incompressible flow over a backward-facing step, Part I: High Reynolds number solutions, Comput. Fluids 37 (2008), pp. 633–655. doi: 10.1016/j.compfluid.2007.09.003
   Web of Science ®Google Scholar
• E. Erturk, T.C. Corke, and C. Gökçöl, Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers, Int. J. Numer. Meth. Fluids 48 (2005), pp. 747–774. doi: 10.1002/fld.953
   Web of Science ®Google Scholar
• C.-M. Fan, C.-H. Yang, and W.-S. Lai, Numerical solutions of two-dimensional flow fields by using the localized method of approximate particular solutions, Eng. Anal. Bound. Elem. 57 (2015), pp. 47–57. doi: 10.1016/j.enganabound.2015.03.012
   Web of Science ®Google Scholar
• A. Fani, S. Camarri, and M.V. Salvetti, Stability analysis and control of the flow in a symmetric channel with a sudden expansion, Phys. Fluids 24 (2012), pp. 1–22. doi: 10.1063/1.4745190
   Web of Science ®Google Scholar
• R.M. Fearn, T. Mullin, and K.A. Cliffe, Nonlinear flow phenomena in a symmetric sudden expansion, J. Fluid Mech. 211 (1990), pp. 595–608. doi: 10.1017/S0022112090001707
   Web of Science ®Google Scholar
• W.F. Florez and H. Power, DRM multidomain mass conservative interpolation approach for the BEM solution of the two-dimensional Navier-Stokes equations, Comput. Math. Appl. 43 (2002), pp. 457–472. doi: 10.1016/S0898-1221(01)00298-X
   Web of Science ®Google Scholar
• A. Fortin, M. Jardak, J. Gervais, and R. Pierre, Localization of Hopf bifurcations in fluid flow problems, Int. J. Numer. Meth. Fluids 24 (1997), pp. 1185–1210. doi: 10.1002/(SICI)1097-0363(19970615)24:11<1185::AID-FLD535>3.0.CO;2-X
   Web of Science ®Google Scholar
• R. Franke, Scattered data interpolation: Test of some methods, Math. Comput. 38 (1982), pp. 181–199.
   Web of Science ®Google Scholar
• U. Ghia, K.N. Ghia, and C.T. Shin, High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method, J. Comput. Phys. 48 (1982), pp. 387–411. doi: 10.1016/0021-9991(82)90058-4
   Web of Science ®Google Scholar
• M.A. Golberg, C.S. Chen, and S.R. Karur, Improved multiquadric approximation for partial differential equations, Eng. Anal. Bound. Elem. 18 (1996), pp. 9–17. doi: 10.1016/S0955-7997(96)00033-1
   Web of Science ®Google Scholar
• M. Golberg and C. Cheng, The method of fundamental solutions for potential, Helmholtz and diffusion problems, in Boundary Integral Methods – Numerical and Mathematical Aspects, M. Golberg, ed., Computational Mechanics Publications, Southampton, 1998, pp. 103–176.
   Google Scholar
• P.M. Gresho, D.K. Gartling, J.R. Torczynski, K.A. Cliffe, K.H. Winters, T.J. Garratt, A. Spence, and J.W. Goodrich, Is the steady viscous incompressible two-dimensional flow over a backward-facing step at Re=800 stable? Int. J. Numer. Methods Fluids 17 (1993), pp. 501–541. doi: 10.1002/fld.1650170605
   Web of Science ®Google Scholar
• Y. Guevel, H. Boutyour, and J.M. Cadou, Automatic detection and branch switching methods for steady bifurcation in fluid mechanics, J. Comput. Phys. 230 (2011), pp. 3614–3629. doi: 10.1016/j.jcp.2011.02.004
   Web of Science ®Google Scholar
• Y. Guevel, G. Girault, and J.M. Cadou, Parametric analysis of steady bifurcations in 2D incompressible viscous flow with high order algorithm, Comput. Fluids 100 (2014), pp. 185–195. doi: 10.1016/j.compfluid.2014.04.032
   Web of Science ®Google Scholar
• J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Springer Netherlands, The Hague, 1983.
   Google Scholar
• R.L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res. 76 (1971), pp. 1905–1915. doi: 10.1029/JB076i008p01905
   Web of Science ®Google Scholar
• S. Hou, Q. Zou, S. Chen, G. Doolen, and A. Cogley, Simulation of cavity flow by the lattice Boltzmann method, J. Comput. Phys. 118 (1995), pp. 329–347. doi: 10.1006/jcph.1995.1103
   Web of Science ®Google Scholar
• L. Kaiktsis, G. Em Karniadakis, and S.A. Orszag, Unsteadiness and convective instabilities in two-dimensional flow over a backward-facing step, J. Fluid Mech. 321 (1996), pp. 157–187. doi: 10.1017/S0022112096007689
   Web of Science ®Google Scholar
• L. Kaiktsis, G.E. Karniadakis, and S.A. Orszag, Onset of three-dimensionality, equilibria, and early transition in flow over a backward-facing step, J. Fluid Mech. 231 (1991), pp. 501–528. doi: 10.1017/S0022112091003488
   Web of Science ®Google Scholar
• E.J. Kansa, Multiquadrics – a scattered data approximation scheme with applications to computational fluid-dynamics-II solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl. 19 (1990), pp. 147–161. doi: 10.1016/0898-1221(90)90271-K
   Web of Science ®Google Scholar
• H.C. Kuhlmann, M. Wanschura, and H.J. Rath, Flow in two-sided lid-driven cavities: Non-uniqueness, instabilities, and cellular structures, J. Fluid Mech. 336 (1997), pp. 267–299. doi: 10.1017/S0022112096004727
   Web of Science ®Google Scholar
• H. Le and P. Moin, An improvement of fractional step methods for the incompressible Navier-Stokes equations, J. Comput. Phys. 92 (1991), pp. 369–379. doi: 10.1016/0021-9991(91)90215-7
   Web of Science ®Google Scholar
• C.K. Lee, X. Liu, and S.C. Fan, Local multiquadric approximation for solving boundary value problems, Comput. Mech. 30 (2003), pp. 396–409. doi: 10.1007/s00466-003-0416-5
   Web of Science ®Google Scholar
• C.Y. Lin, M.H. Gu, D.L. Young, J. Sladek, and V. Sladek, The localized method of approximated particular solutions for solving two-dimensional incompressible viscous flow field, Eng. Anal. Bound. Elem. 57 (2015), pp. 23–36. doi: 10.1016/j.enganabound.2014.11.035
   Web of Science ®Google Scholar
• N. Mai-Duy and T. Tran-Cong, Numerical solution of differential equations using multiquadric radial basis function networks, Neural Netw. 14 (2001), pp. 185–199. doi: 10.1016/S0893-6080(00)00095-2
   PubMed Web of Science ®Google Scholar
• N. Mai-Duy and T. Tran-Cong, Approximation of function and its derivatives using radial basis function networks, Appl. Math. Model. 27 (2003), pp. 197–220. doi: 10.1016/S0307-904X(02)00101-4
   Web of Science ®Google Scholar
• M. Milroy, G. Vickers, and C. Bradley, An adaptive radial basis function approach to modelling scattered data, Int. J. Appl. Sci. Comput. 1 (1994), pp. 319–349.
   Google Scholar
• V.P. Nguyen, T. Rabczuk, S. Bordas, and M. Duflot, Meshless methods: A review and computer implementation aspects, Math. Comput. Simul. 79 (2008), pp. 763–813. doi: 10.1016/j.matcom.2008.01.003
   Web of Science ®Google Scholar
• H. Power and L.C. Wrobel, Boundary Integral Methods in Fluid Mechanics, Computational Mechanics, Southampton, 1995.
   Google Scholar
• H.P. Rani and T.W.H. Sheu, Nonlinear dynamics in a backward-facing step flow, Phys. Fluids 18 (2006), pp. 1–14. doi: 10.1063/1.2261852
   Web of Science ®Google Scholar
• B. Sanderse and B. Koren, Accuracy analysis of explicit Runge–Kutta methods applied to the incompressible Navier–Stokes equations, J. Comput. Phys. 231 (2012), pp. 3041–3063. doi: 10.1016/j.jcp.2011.11.028
   Web of Science ®Google Scholar
• R. Schaback, Multivariate interpolation and approximation by translates of a basis function, in Approximation Theory VIII Approximation and Interpolation, L.L. Schumaker and C. Chui, eds., World Scientific Publishing Co. Inc., Hackensack, 1995, pp. 1–8.
   Google Scholar
• A. Shah, L. Yuan, and S. Islam, Numerical solution of unsteady Navier–Stokes equations on curvilinear meshes, Comput. Math. Appl. 63 (2012), pp. 1548–1556. doi: 10.1016/j.camwa.2012.03.047
   Web of Science ®Google Scholar
• P. Shah, B. Rovagnati, F. Mashayek, and G.B. Jacobs, Subsonic compressible flow in two-sided lid-driven cavity. Part I: Equal walls temperatures, Int. J. Heat Mass Transfer. 50 (2007), pp. 4206–4218. doi: 10.1016/j.ijheatmasstransfer.2007.02.028
   Web of Science ®Google Scholar
• J. Sladek, V. Sladek, C. Tan, and S. Atluri, Analysis of transient heat conduction in 3D anisotropic functionally graded solids, by the MLPG method, CMES 32 (2008), pp. 161–174.
   Web of Science ®Google Scholar
• D. Stevens, H. Power, M. Lees, and H. Morvan, The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems, J. Comput. Phys. 228 (2009), pp. 4606–4624. doi: 10.1016/j.jcp.2009.03.025
   Web of Science ®Google Scholar
• D. Stevens, H. Power, C.Y. Meng, D. Howard, and K.A. Cliffe, An alternative local collocation strategy for high-convergence meshless PDE solutions, using radial basis functions, J. Comput. Phys. 254 (2013), pp. 52–75. doi: 10.1016/j.jcp.2013.07.026
   Web of Science ®Google Scholar
• A. Tarwater, A parametric study of Hardy's multiquadric equations for scattered data interpolation, Tech. Rep. UCRL-54670, Lawrence Livermore National Laboratory, 1985.
   Google Scholar
• E.M. Wahba, Iterative solvers and inflow boundary conditions for plane sudden expansion flows, Appl. Math. Model. 31 (2007), pp. 2553–2563. doi: 10.1016/j.apm.2006.10.017
   Web of Science ®Google Scholar
• F. Wang and A. Rizwan-uddin, A modified nodal scheme for the time-dependent, incompressible Navier-Stokes equations, J. Comput. Phys. 187 (2003), pp. 168–196. doi: 10.1016/S0021-9991(03)00093-7
   Web of Science ®Google Scholar
• G.B. Wright and B. Fornberg, Scattered node compact finite difference-type formulas generated from radial basis functions, J. Comput. Phys. 212 (2006), pp. 99–123. doi: 10.1016/j.jcp.2005.05.030
   Web of Science ®Google Scholar
• G. Yao, J. Kolibal, and C.S. Chen, A localized approach for the method of approximate particular solutions, Comput. Math. Appl. 61 (2011), pp. 2376–2387. doi: 10.1016/j.camwa.2011.02.007
   Web of Science ®Google Scholar
• G. Yao, B. Sarler, and C.S. Chen, A comparison of three explicit local meshless methods using radial basis functions, Eng. Anal. Bound. Elem. 35 (2011), pp. 600–609. doi: 10.1016/j.enganabound.2010.06.022
   Web of Science ®Google Scholar
• K. Zamzamian and M.Y. Hashemi, A novel meshless method for incompressible flow calculations, Eng. Anal. Bound. Elem. 56 (2015), pp. 106–118. doi: 10.1016/j.enganabound.2015.02.009
   Web of Science ®Google Scholar