Solving the stationary Navier-Stokes equations by using Taylor meshless method. (English) Zbl 1404.76197
Summary: The numerical implementation of the Taylor Meshless Method (TMM) for solving the two-dimensional Navier-Stokes equations is considered in this paper, where the TMM is a true meshless integration-free method based on Taylor series. The performance of the proposed numerical technique is evaluated by three benchmark tests. In the first case, the exact solution ia a high degree polynomial that permits to check the consistency and high accuracy of the proposed numerical method. Then the comparison with the analytical solution of the Kovasznay flow for the Reynolds number in the range \([0,1000]\) is considered. It turns out that the TMM works well for large values of Reynolds number and it converges rapidly with the degree of Taylor series (\(p\)-convergence) and the number of sub-domains (\(h\)-convergence). Finally the square lid-driven cavity flow problems for the Reynolds number up to \(\mathrm{Re}=1000\) are solved and excellent agreements with corresponding benchmark solutions are found.
MSC:
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
| 65N99 | Numerical methods for partial differential equations, boundary value problems |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
Software:
LieMathReferences:
| [1] | Zézé, D. S.; Potier-Ferry, M.; Damil, N., A boundary meshless method with shape functions computed from the PDE, Eng Anal Bound Elem, 34, 8, 747-754, (2010) · Zbl 1244.65197 |
| [2] | Tampango, Y.; Potier-Ferry, M.; Koutsawa, Y.; Tiem, S., Coupling of polynomial approximations with application to a boundary meshless method, Int J Numer Meth Eng, 95, 13, 1094-1112, (2013) · Zbl 1352.65598 |
| [3] | Yang, J.; Hu, H.; Potier-Ferry, M., Solving large scale problems by Taylor meshless method, Int J Numer Meth Eng, 112, 103-124, (2017) · Zbl 07867134 |
| [4] | Yang, J.; Hu, H.; Koutsawa, Y.; Potier-Ferry, M., Taylor meshless method for solving non-linear partial differential equations, J Comput Phys, 348, 385-400, (2017) · Zbl 1380.65393 |
| [5] | Tampango, Y.; Potier-Ferry, M.; Koutsawa, Y.; Belouettar, S., Convergence analysis and detection of singularities within a boundary meshless method based on Taylor series, Eng Anal Bound Elem, 36, 10, 1465-1472, (2012) · Zbl 1352.65438 |
| [6] | Kita, E.; Kamiya, N., Trefftz method: an overview, Adv Eng Softw, 24, 3-12, (1995) · Zbl 0984.65502 |
| [7] | Jirousek, J.; Zieliński, A. P., Survey of Trefftz-type element formulations, Comput Struct, 63, 225-242, (1997) · Zbl 0926.74116 |
| [8] | Alves, C. J., On the choice of source points in the method of fundamental solutions, Eng Anal Bound Elem, 33, 1348-1361, (2009) · Zbl 1244.65216 |
| [9] | Antunes, P. R.S., Reducing the ill conditioning in the method of fundamental solutions, Adv Comput Math, 44, 1, 351-365, (2018) · Zbl 1382.65458 |
| [10] | Chu, F.; Wang, L.; Zhong, Z., Finite subdomain radial basis collocation method, Comput Mech, 54, 235-254, (2014) · Zbl 1398.65315 |
| [11] | Balakrishnan, K.; Ramachandran, P. A., Osculatory interpolation in the method of fundamental solution for nonlinear poisson problems, J Comput Phys, 172, 1, 1-18, (2001) · Zbl 0992.65131 |
| [12] | Griewank, A.; Walther, A., Evaluating derivatives: principles and techniques of algorithmic differentiation, (2008), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia · Zbl 1159.65026 |
| [13] | Tian, H.; Potier-Ferry, M.; Abed-Meraim, F., A numerical method based on Taylor series for bifurcation analyses within Föppl-von Karman plate theory, Mech Res Commun, (2018) |
| [14] | Nath, D.; Kalra, M. S.; Munshi, P., One-stage Method of Fundamental and Particular Solutions (MFS-MPS) for the steady Navier-Stokes equations in a lid-driven cavity, Eng Anal Bound Elem, 58, 39-47, (2015) · Zbl 1403.76048 |
| [15] | Bustamante, C. A.; Power, H.; Florez, W. F., A global meshless collocation particular solution method for solving the two-dimensional Navier-Stokes system of equations, Comput Math Appl, 65, 1939-1955, (2013) · Zbl 1383.76366 |
| [16] | Bustamante, C. A.; Power, H.; Florez, W. F., Schwarz alternating domain decomposition approach for the solution of two-dimensional Navier-Stokes flow problems by the method of approximate particular solutions, Numer Meth Part D E, 31, 777-797, (2015) · Zbl 1325.65172 |
| [17] | Granados, J. M.; Power, H.; Bustamante, C. A.; Flórez, W. F.; Hill, A. F., A global particular solution meshless approach for the four-sided lid-driven cavity flow problem in the presence of magnetic fields, Comput Fluids, 160, 120-137, (2018) · Zbl 1390.76673 |
| [18] | Chen, C. S.; Fan, C. M.; Wen, P. H., The method of approximate particular solutions for solving certain partial differential equations, Numer Meth Part D E, 28, 506-522, (2012) · Zbl 1242.65267 |
| [19] | Chinchapatnam, P. P.; Djidjeli, K.; Nair, P. B., Radial basis function meshless method for the steady incompressible Navier-Stokes equations, Int J Comput Math, 84, 1509-1521, (2007) · Zbl 1123.76048 |
| [20] | Poitou, A.; Bouberbachene, M.; Hochard, C., Resolution of three-dimensional Stokes fluid flows using a Trefftz method, Comput Method Appl M, 190, 561-578, (2000) · Zbl 0993.76065 |
| [21] | Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv Comput Math, 9, 1-2, 69-95, (1998) · Zbl 0922.65074 |
| [22] | Zhang, X.; Liu, X. H.; Song, K. Z.; Lu, M. W., Least-squares collocation meshless method, Int J Numer Meth Eng, 51, 9, 1089-1100, (2001) · Zbl 1056.74064 |
| [23] | Babuška, I.; Banerjee, U., Stable generalized finite element method (SGFEM), Comput Method Appl M, 201, 91-111, (2012) · Zbl 1239.74093 |
| [24] | Liu, X.; Li, J.; Chen, Z., A weak Galerkin finite element method for the Navier-Stokes equations, J Comput Appl Math, 333, 442-457, (2018) · Zbl 1395.76040 |
| [25] | Muller, J.-M.; Brisebarre, N.; de Dinechin, F.; Jeannerod, C.-P.; Lefèvre, V.; Melquiond, G., Handbook of floating-point arithmetic, (2010), Birkhäuser Basel · Zbl 1197.65001 |
| [26] | Kansa, E. J.; Holoborodko, P., On the ill-conditioned nature of C^{∞} RBF strong collocation, Eng Anal Bound Elem, 78, 26-30, (2017) · Zbl 1403.65160 |
| [27] | Kovasznay, L. I.G., Laminar flow behind a two-dimensional grid, Math Proc Cambridge, 44, 1, 58-62, (1948) · Zbl 0030.22902 |
| [28] | Schaback, R., Error estimates and condition numbers for radial basis function interpolation, Adv Comput Math, 3, 3, 251-264, (1995) · Zbl 0861.65007 |
| [29] | Ghia, U.; Ghia, K. N.; Shin, C., High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J Comput Phys, 48, 3, 387-411, (1982) · Zbl 0511.76031 |
| [30] | Cadou, J.; Potier-Ferry, M.; Cochelin, B.; Damil, N., ANM for stationary Navier-Stokes equations and with Petrov-Galerkin formulation, Int J Numer Meth Eng, 50, 4, 825-845, (2001) · Zbl 1013.76046 |
| [31] | Ning, Y.; Premnath, K. N.; Patil, D. V., Numerical study of the properties of the central moment lattice Boltzmann method, Int J Numer Meth Fl, 82, 2, 59-90, (2016) |
| [32] | Shao, Q.; Fahs, M.; Younes, A.; Makradi, A.; Mara, T., A new benchmark reference solution for double-diffusive convection in a heterogeneous porous medium, Numer Heat Tr B-Fund, 70, 5, 373-392, (2016) |
| [33] | Medale, M.; Cochelin, B., A parallel computer implementation of the asymptotic numerical method to study thermal convection instabilities, J Comput Phys, 228, 22, 8249-8262, (2009) · Zbl 1422.76149 |
| [34] | Bischof, C.; Corliss, G.; Griewank, A., Structured second-and higher-order derivatives through univariate Taylor series, Optim Method Softw, 2, 3-4, 211-232, (1993) |
| [35] | Griewank, A.; Utke, J.; Walther, A., Evaluating higher derivative tensors by forward propagation of univariate Taylor series, Math Comput, 69, 231, 1117-1130, (2000) · Zbl 0952.65028 |
| [36] | Neidinger, R. D., Directions for computing truncated multivariate Taylor series, Math Comput, 74, 249, 321-340, (2005) · Zbl 1052.65012 |
| [37] | Neidinger, R. D., Efficient recurrence relations for univariate and multivariate Taylor series coefficients, J Am Inst Math Sci, 587-596, (2013) · Zbl 1307.65027 |
| [38] | Kaltchev, D., Building truncated Taylor maps with mathematica and applications to FFAG, Proceedings of the 9th European Particle Accelerator Conference, 1822-1824, (2004), Lucerne, Switzerland |
| [39] | Kaltchev, D., Implementation of TPSA in the mathematica code LieMath, Proceedings of the 10th European Particle Accelerator Conference, 2077-2079, (2006), Edinburgh, UK |
| [40] | Kaltchev, D., Mathematica program for extracting one-turn lie generator map application of TPSA, Phys Proc, 1, 1, 333-338, (2008) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
