Proposta de um procedimento híbrido para estimar e reduzir o erro de iteração em problemas de transferência de calor - Dall'Agnol, Caroline (2020)
This study aims to improve the techniques for estimates of iteration errors in heat transfer problems. First, a new estimator of iteration error is proposed and, based on the estimates obtained, a method is proposed to improve the iteration error predictions in ranges of iterations where the estimator does not present accurate results. The proposed estimator is an empirical estimator that provides iteration errors estimates based on interest variables convergence rate. Its performance was tested in two one-dimensional equations: Poisson’s equation and advection-diffusion equation, and in a two-dimensional equation: Laplace’s equation. All equations were discretized using the Finite Difference Method (FDM) in uniform meshes. The systems of equations resulting from the discretizations were solved by the TriDiagonal Matrix Algorithm solver (TDMA), PentaDiagonal Matrix Algorithm solver (PDMA) and Gauss-Seidel solver (GS). The TDMA solver was used to obtain the one-dimensional equations direct solution, the PDMA solver to obtain the bidimensional equation reference solution and the GS solver to estimate errors at each iteration. The Laplace equation was solved with and without the multigrid method associated with GS to accelerate convergence. The variables chosen to evaluate the results were: the function value at the central point of the domain (local), the derivative at the right boundary (local) and the function mean value (global). The codes were implemented in Fortran 95 language, with quadruple precision, in the Microsoft Visual Studio Community 2013. The proposed estimator was evaluated with respect to its accuracy and reliability and was compared to the main iteration error estimators found in the literature and it was observed that, for all problems and variables, the estimators have similar results. The iterations initial ranges and the final ranges were those that presented the least accurate estimates. Thus, in order to improve the estimates obtained, first it was delimited, for each variable, the iterations ranges in which the estimator presents the best error estimates. To that end, the criteria were: geometric series convergence that represents the proposed estimator, monotonic convergence and the interference of round-off errors. After identifying the interval with the best estimates, these same estimates were used to obtain solutions with reduced iteration errors. The best estimate range last solution with reduced iteration errors was used to recalculate the predictions. Through this procedure it was possible to improve the iteration initial range estimates. Combining the error estimates with the estimator and those improved by the proposed method, a hybrid procedure is obtained to estimate the iteration error throughout the iterative cycle.