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A novel upwind-based local radial basis function differential quadrature method for convection-dominated flows. (English) Zbl 1391.76529
Summary: In this paper, a new upwind technique for local radial basis function differential quadrature (LRBF-DQ) method is proposed to solve the convection-dominated flow problems. By using a modified Euclidean distance function according to the local flow direction and the value of parameter that controls the convection effect, the local support in the formulation of LRBF-DQ can be chosen in a way shifting towards the upstream direction to form a comet-like shape. The upwind effect is therefore naturally incorporated when computing the weighting coefficients for LRBF-DQ method. The capability of the proposed method is examined by solving two-dimensional convection-diffusion equation with various Peclet numbers and magnetohydrodynamics (MHD) problems with very high Hartmann numbers. The results show that remarkable improvement of accuracy can be achieved by the current upwind-based LRBF-DQ method than the conventional ones.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
65N99 Numerical methods for partial differential equations, boundary value problems
76W05 Magnetohydrodynamics and electrohydrodynamics
Software:
Mfree2D
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[1] Versteeg, H. K.; Malalasekera, W., An introduction to computational fluid dynamics: the finite volume method, (2007), Prentice Hall
[2] Courant, R.; Isaacson, E.; Rees, M., On the solution of nonlinear hyperbolic differential equations by finite differences, Commun Pur Appl Math, 5, 3, 243-255, (1952) · Zbl 0047.11704
[3] Warming, R. F.; Beam, R. M., Upwind second-order difference schemes and applications in aerodynamic flows, AIAA J, 14, 1241-1249, (1976) · Zbl 0364.76047
[4] Leonard, B., A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Comput Methods Appl Mech Eng, 19, 1, 59-98, (1979) · Zbl 0423.76070
[5] Brooks, A. N.; Hughes, T. J.R., Streamline upwind Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput Methods Appl Mech Eng, 32, 1-3, 199-259, (1982) · Zbl 0497.76041
[6] Whiting, C. H.; Jansen, K. E., A stabilized finite element method for the incompressible Navier-Stokes equations using a hierarchical basis, Int J Numer Methods Fluids, 35, 1, 93-116, (2001) · Zbl 0990.76048
[7] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput Methods Appl Mech Eng, 139, 1, 3-47, (1996) · Zbl 0891.73075
[8] Liu, G. R., Mesh free methods: moving beyond the finite element method, (2002), CRC Press Boca Raton, FL
[9] Nguyen, V. P.; Rabczuk, T.; Bordas, S.; Duflot, M., Meshless methods: a review and computer implementation aspects, Math Comput Simulat, 79, 3, 763-813, (2008) · Zbl 1152.74055
[10] Shu, C.; Ding, H.; Yeo, K., Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations, Comput Methods Appl Mech Eng, 192, 7, 941-954, (2003) · Zbl 1025.76036
[11] Buhmann, M. D., Radial basis functions: theory and implementations, (2003), Cambridge University Press · Zbl 1038.41001
[12] Bellman, R. E.; Casti, J., Differential quadrature and long-term integration, J Math Anal Appl, 34, 235-238, (1971) · Zbl 0236.65020
[13] Bellman, R.; Kashef, B.; Casti, J., Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, J Comput Phys, 10, 1, 40-52, (1972) · Zbl 0247.65061
[14] Shu, C.; Ding, H.; Chen, H.; Wang, T., An upwind local RBF-DQ method for simulation of inviscid compressible flows, Comput Methods Appl Mech Eng, 194, 18-20, 2001-2017, (2005) · Zbl 1093.76052
[15] Ding, H.; Shu, C.; Yeo, K.; Lu, Z., Simulation of natural convection in eccentric annuli between a square outer cylinder and a circular inner cylinder using local MQ-DQ method, Numer Heat Tr A-Appl, 47, 3, 291-313, (2005)
[16] Ding, H.; Shu, C.; Yeo, K.; Xu, D., Numerical computation of three-dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method, Comput Methods Appl Mech Eng, 195, 7-8, 516-533, (2006) · Zbl 1222.76072
[17] Shan, Y.; Shu, C.; Lu, Z., Application of local MQ-DQ method to solve 3D incompressible viscous flows with curved boundary, CMES-Comput Model Eng, 25, 2, 99-113, (2008)
[18] Khoshfetrat, A.; Abedini, M., Numerical modeling of long waves in shallow water using LRBF-DQ and hybrid DQ/LRBF-DQ, Ocean Model, 65, 0, 1-10, (2013)
[19] Wu, W.; Shu, C.; Wang, C., Vibration analysis of arbitrarily shaped membranes using local radial basis function-based differential quadrature method, J Sound Vib, 306, 1-2, 252-270, (2007)
[20] Soleimani, S.; Jalaal, M.; Bararnia, H.; Ghasemi, E.; Ganji, D.; Mohammadi, F., Local RBF-DQ method for two-dimensional transient heat conduction problems, Int Commun Heat Mass, 37, 9, 1411-1418, (2010)
[21] Soleimani, S.; Ganji, D.; Ghasemi, E.; Jalaal, M.; Bararnia, H., Meshless local RBF-DQ for 2-D heat conduction: a comparative study, Therm Sci, 15, suppl., S117-S121, (2011)
[22] Demendy, Z.; Nagy, T., A new algorithm for solution of equations of MHD channel flows at moderate Hartmann numbers, Acta Mech, 123, 1-4, 135-149, (1997) · Zbl 0902.76058
[23] Nesliturk, A.; Tezer-Sezgin, M., The finite element method for MHD flow at high Hartmann numbers, Comput Methods Appl Mech Eng, 194, 9-11, 1201-1224, (2005) · Zbl 1091.76036
[24] Verardi, J. C.S. L.L.; Machado, J. M., The element-free Galerkin method applied to the study of fully developed magnetohydrodynamic duck flows, IEEE Trans Magn, 38, 941-944, (2002)
[25] Zhang, L.; Ouyang, J.; Zhang, X., The two-level element free Galerkin method for MHD flow at high Hartmann numbers, Phys Lett A, 372, 5625-5638, (2008) · Zbl 1223.76128
[26] Li, Y.; Tian, Z. F., An exponential compact difference scheme for solving 2D steady magnetohydrodynamic (MHD) duct flow problems, J Comput Phys, 231, 16, 5443-5468, (2012) · Zbl 1431.76096
[27] Cai, X.; Su, G. H.; Qiu, S., Upwinding meshfree point collocation method for steady MHD flow with arbitrary orientation of applied magnetic field at high Hartmann numbers, Comput Fluids, 44, 1, 153-161, (2011) · Zbl 1271.76239
[28] Tezer-Sezgin, M., Solution of magnetohydrodynamic flow in a rectangular duct by differential quadrature method, Comput Fluids, 33, 4, 533-547, (2004) · Zbl 1137.76453
[29] Ögˇüt, E., Magnetohydrodynamic natural convection flow in an enclosure with a finite length heater using the differential quadrature (DQ) method, Numer Heat Tr A-Appl, 58, 11, 900-921, (2010)
[30] Hardy, R. L., Multiquadric equations of topography and other irregular surfaces, J Geophys Res, 76, 1905-1915, (1971)
[31] Franke, R., Scattered data interpolation: tests of some method, Math Comput, 38, 157, 181-200, (1982) · Zbl 0476.65005
[32] Barrett, K. E., Duct flow with a transverse magnetic field at high Hartmann numbers, Int J Numer Meth Eng, 50, 8, 1893-1906, (2001) · Zbl 0998.76045
[33] Shercliff, J. A.; Batchelor, G. K., Steady motion of conducting fluids in pipes under transverse magnetic fields, Proc Camb Philos Soc, 49, 136-144, (1953) · Zbl 0050.19404
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