Comparison between extrapolation techniques associated to the multigrid method applied in the Navier-Stokes equations
Bruno Benato Rutyna1; Marcio Augusto Villela Pinto1; Réverton Luís Antunes Neundorf1; Márcio Alexandro Maciel de Anunciação1; Márcio André Martins2
1 Federal University of Paraná; 2 State University of Centro-Oeste
doi:10.20906/CPS/CILAMCE2017-0142
Resumo
Among the problems of Computational Fluid Dynamics (CFD), the one that deserves attention is finding satisfactory solutions for the Navier-Stokes equations, which in this work they represent a case of two-dimensional laminar flow of a time-dependent incompressible fluid. Given the difficulty to obtain the analytical solutions, to use iterative computational methods is a feasible option. In this work it was decided to solve them iteratively through the projection method. Nevertheless, the approximations obtained during the iterations can converge very slowly, thus consuming an undesired computational time. Therefore, it becomes useful to use the multigrid method to solve these systems of equations originated from the discretized problem, which is a powerful convergence accelerator and has low computational cost. Further gain can be achieved by applying solution extrapolation techniques, which can be used after the multigrid cycles or adapted to intercalate these cycles. This work will first analyze the application of extrapolators in the post-multigrid case, taking into account the following parameters: CPU time, storage memory peak, dimensionless norm of the residual based on the initial estimate, convergence factors and error standards of iteration. The extrapolator that obtains the best performance will be used in the second stage of this work, which consists in measuring again the mentioned parameters, but with the extrapolator applied between the cycles of the multigrid method. The extrapolation methods used are: Aitken, Empiric, Mitin, Scalar epsilon, Scalar rho, Tophological epsilon and Tophological rho. In the end, it is expected to be able to affirm which extrapolator is most effective and also if its use overcomes the use of the multigrid method without the use of extrapolation techniques.
Palavras-chave: multigrid method; extrapolation methods; Navier-Stokes; convergence acceleration; Computational Fluid Dynamics (CFD)