Accommodative FAS-FMG Multilevel Based Meshfree Augmented RBF-FD Method for Navier-Stokes Equations in Spherical Geometr
The efficiency of any numerical scheme measures on the accuracy of the scheme and its computational time. An efficient meshfree augmented local radial basis function (RBF-FD) method has been developed for steady incompressible Navier-Stokes equations in s
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Abstract. The efficiency of any numerical scheme measures on the accuracy of the scheme and its computational time. An efficient meshfree augmented local radial basis function (RBF-FD) method has been developed for steady incompressible Navier-Stokes equations in spherical geometry with unbounded domain which is based on accommodative FAS-FMG multigrid method. The axi-symmetric spherical polar NavierStokes equations are solved without using transformation. The non-linear convective terms are handled efficiently by considering upwind type of RBF nodes. The developed scheme saves around 34% of the CPU time than the usual RBF-FD method. Keywords: Radial basis function · Accommodative FAS-FMG multilevel method · Meshless method · Unbounded flows · Navier-Stokes equations
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Introduction
The increasing use of computational fluid dynamics (CFD) for engineering design and analysis demands highly efficient solution methods. The discretization of numerical methods for solving elliptic Navier-Stokes(N-S) equations generally results in solving a system of algebraic equations. If the number of unknowns are large, solving by a direct method, such as Gaussian elimination, can be inefficient. Therefore, iterative methods like point Gauss-Seidel and line Gauss-Seidel are used to solve the huge linearized system of equations. For better convergence of the iterative methods, a good initial solution is essential. It was also found that Gauss-Seidel iterative method is effective for the first few iterations and then the error elimination process becomes slow. Based on this fact, a fast finite difference numerical method has been developed by Hyman [1] to solve elliptic partial differential equations with Dirichlet boundary conditions. His method is based on a local mesh refinement technique which provides a better initial guess for the iterative algorithms. The solution is achieved quickly and the CPU c Springer Nature Singapore Pte Ltd. 2017 D. Giri et al. (Eds.): ICMC 2017, CCIS 655, pp. 141–151, 2017. DOI: 10.1007/978-981-10-4642-1 13
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N.B. Barik and T.V.S. Sekhar
time is minimized. Over the past few decades, finite difference based multigrid methods have been developed to solve the system of equations so as to improve the convergence rate of iterative methods and hence their efficiency. Ghia et al. [2] developed accommodative version of the Full Approximation Scheme-Full MultiGrid (FAS-FMG) procedure of Brandt [3] and applied this to Navier-Stokes equations. It is well known that RBF based methods suffer from high computational cost compared to conventional mesh based methods. The calculation of RBF weights corresponding to the neighboring particles of a data point, requires expansive square root and matrix inversion processes. Moreover, the calculation of derivative approximation at a given order of accuracy usually requires more number of neighboring particles (or nodes) for meshfree methods in an irregular grid than for finite difference method (FDM) on a cartesian grid. As a result, the bandwidth of matrices representing the governing algeb
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