An Efficient Local RBF Meshless Scheme for Steady Convection-Diffusion Problems

The efficiency of any numerical scheme measures on the accuracy of the scheme and its computational time. In this work, an efficient augmented local radial basis function finite difference (RBF-FD) scheme has been developed for steady convection-diffusion problems. The accommodative full approximation scheme-full multigrid (FAS-FMG) analogy with local refinement is adopted to achieve this efficiency. The efficiency of the method is illustrated through linear convection-diffusion equation, coupled nonlinear equation and for the incompressible Navier-Stokes (NS) equations. Numerical studies are made by using multiquadric (MQ) RBF. The upwind type nodes are considered to handle convective terms effectively for NS equations. The developed scheme saves 70-80% of the CPU time for the first two model problems and at least 50% of the CPU time for NS equations than the usual RBF-FD method found in the literature. It is found that there are optimum sets of nodes which give accurate and stable solution. It is also found that the value of the shape parameter will be lower when the number of nodes is higher. But higher the value of Re, higher will be the shape parameter.