zbMATH — the first resource for mathematics

Development of least-square-based two-dimensional finite difference schemes and their application to simulate natural convection in a cavity. (English) Zbl 1033.76039
Summary: We present two-dimensional mesh-free finite difference schemes for solving incompressible viscous flows. The method is based on the use of a weighted least-square approximation procedure together with Taylor series expansion of unknown functions. Discretization error for derivatives is investigated analytically on uniform mesh, and convergence property of the method is numerically tested. The role of the weighting function in the method is studied. Neumann-type boundary condition is treated by applying locally orthogonal boundary grids. Application to a natural convection in cavity is demonstrated on three different types of point distribution.

76M20 Finite difference methods applied to problems in fluid mechanics
76R10 Free convection
80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text: DOI
[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. eng., 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. eng., 40, 727-758, (1997) · Zbl 0949.65117
[6] Duarte CA, Oden JT. Hp clouds–a meshless method to solve boundary-value problems. TICAM Report 95-05
[7] Oñate, 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. eng., 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. eng., 20, 1599-1612, (1984) · Zbl 0544.65006
[10] Schnauer, W.; Adolph, T., How we solve pdes, J. comput. appl. math., 131, 473-492, (2001) · Zbl 0982.65113
[11] Black, T.; Belytschko, T., Convergence of corrected derivative methods for second-order linear partial differential equations, Int. J. numer. meth. eng., 44, 177-203, (1999) · Zbl 0938.65134
[12] Nair, M.T.; Sengupta, T.K., Orthogonal grid generation for navier – stokes computations, Int. J. numer. meth. fluids, 28, 215-224, (1998) · Zbl 0913.76059
[13] 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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.