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Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations. (English) Zbl 1025.76036
Summary: Local radial basis function-based differential quadrature method is presented in detail in this paper. The method is a natural mesh-free approach. Like the conventional differential quadrature (DQ) method, it discretizes any derivative at a knot by a weighted linear sum of functional values at its neighbouring knots, which may be distributed randomly. However, different from the conventional DQ method, the weighting coefficients in present method are determined by taking the radial basis functions (RBFs) instead of high order polynomials as the test functions. The method works in a similar fashion as conventional finite difference schemes but with ”truly” mesh-free property. In this paper, we mainly concentrate on the multiquadric RBFs since they have exponential convergence. The effects of shape parameter c on the accuracy of numerical solution of linear and nonlinear partial differential equations are studied, and how the value of optimal c varies with the number of local support knots is also numerically demonstrated. The proposed method is validated by its application to the simulation of natural convection in a square cavity. Excellent numerical results are obtained on an irregular knot distribution.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Lucy, L.B, A numerical approach to the testing of the fission hypothesis, Astro. J., 8, 1013-1024, (1977)
[2] Nayroles, B; Touzot, G; Villon, P, Generalizing the finite element method: diffuse approximation and diffuse elements, Comput. mech., 10, 307-318, (1992) · Zbl 0764.65068
[3] Belytschko, T; Lu, Y.Y; Gu, L, Element-free Galerkin methods, Int. J. numer. meth. engrg., 37, 229-256, (1994) · Zbl 0796.73077
[4] Liu, W; Jun, S; Zhang, Y, Reproducing kernel particle methods, Int. J. numer. meth. fluids, 20, 1081-1106, (1995) · Zbl 0881.76072
[5] Babuska, I; Melenk, J, The partition of unity method, Int. J. numer. meth. engrg., 40, 727-758, (1997) · Zbl 0949.65117
[6] C.A. Duarte, J.T. Oden, Hp clouds–a meshless method to solve boundary-value problems, TICAM Report 95-05
[7] Ońate, E; Idelsohn, S; Zienkiewicz, O.C; Taylor, R.L, A finite point method in computational mechanics, application to convective transport and fluid flow, Int. J. numer. meth. engrg., 39, 3839-3866, (1996) · Zbl 0884.76068
[8] Atluri, S.N; Zhu, T, New meshless local petrov – galerkin (MLPG) approach in computational mechanics, Comput. mech., 22, 2, 117-127, (1998) · Zbl 0932.76067
[9] Liszka, T, An interpolation method for an irregular net of nodes, Int. J. numer. meth. engrg., 20, 1599-1612, (1984) · Zbl 0544.65006
[10] Kansa, E.J, Multiquadrics–A scattered scattered data approximation scheme with applications to computational fluid-dynamics–I. surface approximations and partial derivative estimates, Comput. math. appl., 19, 6-8, 127-145, (1990) · Zbl 0692.76003
[11] Kansa, E.J, Multiquadrics–A scattered scattered data approximation scheme with applications to computational fluid-dynamics–II. solutions to parabolic, hyperbolic, and elliptic partial differential equations, Comput. math. appl., 19, 6-8, 147-161, (1990) · Zbl 0850.76048
[12] Fasshauer, G.E, Solving partial differential equations by collocation with radial basis functions, (), 131-138 · Zbl 0938.65140
[13] Forberg, B; Driscoll, T.A, Interpolation in the limit of increasingly flat radial basis functions, Comput. math. appl., 43, 413-422, (2002) · Zbl 1006.65013
[14] Hon, Y.C; Wu, Z.M, A quasi-interpolation method for solving stiff ordinary differential equations, Int. J. numer. meth. engrg., 48, 1187-1197, (2000) · Zbl 0962.65060
[15] Chen, W; Tanaka, M, A meshless, integration-free, and boundary-only RBF technique, Comput. math. appl., 43, 379-391, (2002) · Zbl 0999.65142
[16] Chen, C.S; Brebbia, C.A; Power, H, Dual reciprocity method using for Helmholtz-type operators, Boundary elem., 20, 495-504, (1998) · Zbl 0929.65109
[17] Wu, Z.M, Hermite – bikhoff interpolation of scattered data by radial basis function, Approx. theory appl., 8, 1-10, (1992)
[18] Dubal, M.R; Olivera, S.R; Matzner, R.A, Approaches to numerical relativity, (1993), Cambridge University Press Cambridge, UK
[19] Fornberg, B; Driscoll, T.A; Wright, G; Charles, R, Observations on the behavior of radial basis function approximations near boundaries, Comput. math. appl., 43, 473-490, (2002) · Zbl 0999.65005
[20] Kansa, E.J; Hon, Y.C, Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations, Comput. math. appl., 39, 123-137, (2000) · Zbl 0955.65086
[21] Hardy, R.L, Multiquadric equations of topography and other irregular surfaces, J. geophys. res., 76, 1905-1915, (1971)
[22] Franke, R, Scattered data interpolation: tests of some methods, Math. comp., 38, 181-199, (1982) · Zbl 0476.65005
[23] Shu, C, Differential quadrature and its application in engineering, (2000), Springer-Verlag London Limited · Zbl 0944.65107
[24] Shu, C; Richards, B.E, Application of generalized differential quadrature to solve two-dimensional incompressible navier – stokes equations, Int. J. numer. meth. fluids, 15, 791-798, (1992) · Zbl 0762.76085
[25] Shu, C; Chew, Y.T, Fourier expansion-based differential quadrature and its application to Helmholtz eigenvalue problems, Commun. numer. meth. engrg., 13, 643-653, (1997) · Zbl 0886.65109
[26] de Vahl Davis, G, Natural convection of air in a square cavity: a benchmark numerical solution, Int. J. numer. meth. fluids, 3, 249-264, (1983) · Zbl 0538.76075
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