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# High Order of Accuracy for Poisson Equation Obtained by Grouping of Repeated Richardson Extrapolation with Fourth Order Schemes

Luciano Pereira da Silva1,*, Bruno Benato Rutyna1, Aline Roberta Santos Righi2, Marcio Augusto Villela Pinto3

1 Graduate Program in Numerical Methods in Engineering, Federal University of Paraná, Curitiba, 81531-980, Brazil
2 Graduate Program in Space Sciences and Technologies, Aeronautics Institute of Technology, São José dos Campos, 12228-900, Brazil
3 Department of Mechanical Engineering, Federal University of Paraná, Curitiba, 81531-980, Brazil

* Corresponding Author: Luciano Pereira da Silva. Email:

(This article belongs to this Special Issue: Intelligent Computing for Engineering Applications)

Computer Modeling in Engineering & Sciences 2021, 128(2), 699-715. https://doi.org/10.32604/cmes.2021.014239

## Abstract

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.

## Keywords

da Silva, L. P., Rutyna, B. B., Righi,  A. R. S.,  Pinto, M. A. V. (2021). High Order of Accuracy for Poisson Equation Obtained by Grouping of Repeated Richardson Extrapolation with Fourth Order Schemes. CMES-Computer Modeling in Engineering & Sciences, 128(2), 699–715.

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