In this article, we improve the order of precision of the two-dimensional Poisson equation by combining extrapolation techniques with high order schemes. The high order solutions obtained traditionally generate non-sparse matrices and the calculation time is very high. We can obtain sparse matrices by applying compact schemes. In this article, we compare compact and exponential finite difference schemes of fourth order. The numerical solutions are calculated in quadruple precision (Real * 16 or extended precision) in FORTRAN language, and iteratively obtained until reaching the round-off error magnitude around 1.0E−32. This procedure is performed to ensure that there is no iteration error. The Repeated Richardson Extrapolation (RRE) method combines numerical solutions in different grids, determining higher orders of accuracy. The main contribution of this work is based on a process that initializes with fourth order solutions combining with RRE in order to find solutions of sixth, eighth, and tenth order of precision. The multigrid Full Approximation Scheme (FAS) is also applied to accelerate the convergence and obtain the numerical solutions on the fine grids.

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In this work, we applied different extrapolation techniques in association with the multigrid method to discover which one is the most effective in reducing the iteration error and the processing time (CPU time), as well as in improving the convergence factors. The mathematical model studied refers to the two-dimensional laminar flow of an isothermal time-dependent incompressible fluid modeled by the Navier-Stokes equations, with , solved iteratively with the projection method and the Finite Volume Method. The extrapolation methods used were: Aitken, Empiric, Mitin, scalar Epsilon, scalar Rho, topological Epsilon, and topological Rho. A two-step application was performed: first, extrapolators methods were applied individually after the use of the multigrid method. Then, the best-performing extrapolation techniques were used in the second step, where they were applied between the cycles of the multigrid method. The methods that presented the best convergence properties in the first stage were topological and scalar Epsilon. In the second stage, both methods maintained their performance, however, the topological Epsilon method presented more significant convergence rates than the scalar Epsilon. The other parameters analyzed were: the storage memory peak, the dimensionless norm of the residual based on the initial estimate, and the error norms of iteration. Thus, it was possible to state which extrapolation technique performed best and to compare it with the multigrid method with no extrapolation, which in this study was the topological Epsilon method.