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SUMMARYThis paper describes the application of radial basis function (RBF) based finite difference type scheme (RBF-FD) for solving steady convection-diffusion equations. Numerical studies are made using multiquadric (MQ) RBF. By varying the shape parameter in MQ, the accuracy of the solution is seen to be highly improved for large values of Reynolds' numbers. The developed scheme has been compared with the corresponding finite difference scheme and found that the solutions obtained using the former are non-oscillatory.

In the present investigation an attempt has been made to solve the two-dimensional incompressible viscous flow past an impulsively started circular cylinder for Reynolds numbers ranging from 20 to 5000 using higher-order semicompact scheme (HOSC). Unlike conventional higher-order compact schemes the HOSC scheme has been developed to handle the circular geometry of the chosen problem and the intensive algebraic manipulations have been reduced considerably by relaxing the compactness of the computational stencil for few terms (but retained for most of the terms) of the discretized equations. For the flow past an impulsively started circular cylinder the results obtained at low and moderate Reynolds numbers have been validated with the experimental and numerical observations available in the literature. For high Reynolds number flows, the present scheme rightly captures the alpha phenomenon at Re=1000 and both beta and alpha phenomena one after the other at Re=5000 .

Exponential compact higher-order schemes have been developed for unsteady convection-diffusion equation (CDE). One of the developed scheme is sixth-order accurate which is conditionally stable for the Péclet number 0 ≤ Pe ≤ 2.8 and the other is fourth-order accurate which is unconditionally stable. Schemes for two-dimensional (2D) problems are made to use alternate direction implicit (ADI) algorithm. Example problems are solved and the numerical solutions are compared with the analytical solutions for each case.

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SUMMARYIn this paper, two radial basis function (RBF)-based local grid-free upwind schemes have been discussed for convection-diffusion equations. The schemes have been validated over some convection-diffusion problems with sharp boundary layers. It is found that one of the upwind schemes realizes the boundary layers more accurately than the rest. Comparisons with the analytical solutions demonstrate that the local RBF grid-free upwind schemes based on the exact velocity direction are stable and produce accurate results on domains discretized even with scattered distribution of nodal points.

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