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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: email

(This article belongs to the 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.

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APA Style
Silva, L.P.D., 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. Computer Modeling in Engineering & Sciences, 128(2), 699-715. https://doi.org/10.32604/cmes.2021.014239
Vancouver Style
Silva LPD, Rutyna BB, Righi ARS, Pinto MAV. High order of accuracy for poisson equation obtained by grouping of repeated richardson extrapolation with fourth order schemes. Comput Model Eng Sci. 2021;128(2):699-715 https://doi.org/10.32604/cmes.2021.014239
IEEE Style
L.P.D. Silva, B.B. Rutyna, A.R.S. Righi, and M.A.V. Pinto, “High Order of Accuracy for Poisson Equation Obtained by Grouping of Repeated Richardson Extrapolation with Fourth Order Schemes,” Comput. Model. Eng. Sci., vol. 128, no. 2, pp. 699-715, 2021. https://doi.org/10.32604/cmes.2021.014239



cc Copyright © 2021 The Author(s). Published by Tech Science Press.
This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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