In 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. Copyright © 2009 John Wiley & Sons, Ltd.

Sanyasiraju V.S.S. Yedida

Department of Mathematics

Analytical solutions

Convection diffusion

Convection diffusion problems

Convection-diffusion equations

LOCAL GRIDS

MULTIQUADRICS

NODAL POINTS

Radial basis functions

SHARP BOUNDARIES

STEADY CONVECTION

UP-WIND SCHEME

Attitude control

Boundary layers

Diffusion in liquids

Heat convection

Partial differential equations

Radial basis function networks

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IIT Madras

International Journal for Numerical Methods in Fluids

The most common strategy existing in the literature for solving convection dominated Convection-Diffusion Equations (CDE) is using central approximation to the diffusive terms and upwind approximation to the convective terms. In the present work, we propose a multiquadric local RBF based grid-free upwind (LRBF-U) scheme for solving convection dominated CDE. In this method, the entire CDE operator is discretized over the nodes in the upwind local support domain for strongly convection dominant problems. The variable (optimal) shape parameter for LRBF-U scheme has been obtained by using a local optimization algorithm developed by the authors. It has been observed that for highly convection dominated problems, the LRBF-U scheme produces stable and accurate results. The proposed scheme is also been compared with the conventional Central-Upwind combined scheme, to demonstrate its superiority in generating high accurate solutions than the latter. © 2014 Elsevier Ltd.

Engineering Analysis with Boundary Elements

Upwind strategies for local RBF scheme to solve convection dominated problems

A 'local' radial basic function (RBF) based gridfree scheme has been developed to solve unsteady, incompressible Navier-Stokes equations in primitive variables. The velocity-pressure decoupling is obtained by making use of a fractional step algorithm. The scheme is validated over a variety of benchmark problems and found a very good agreement with the existing results. Comparisons with the benchmark solutions show that the developed local RBF gridfree scheme is stable and produces accurate results on domains discretized even with non-uniform distribution of nodal points. © 2008 Elsevier Inc. All rights reserved.

Journal of Computational Physics

Local radial basis function based gridfree scheme for unsteady incompressible viscous flows

This 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. Copyright © 2007 John Wiley & Sons, Ltd.

International Journal for Numerical Methods in Engineering

Local RBF-FD solutions for steady convection-diffusion problems

Computational Mechanics

Meshless techniques for convection dominated problems

Scattered node compact finite difference-type formulas generated from radial basis functions

Computers & Mathematics with Applications

Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations

A note on two upwind strategies for RBF-based grid-free schemes to solve steady convection-diffusion equations

Journal	International Journal for Numerical Methods in Fluids
ISSN	02712091
Open Access	No