Iteration error estimation in Computacional Fluid Dynamics using a new Empirical Estimator

Resumo

This study aims to improve the techniques for estimates of iteration errors, through the use of a
new estimator. The new estimator provides estimates of iteration errors based on variables convergence rate.
Its performance was tested in two one-dimensional steady-state mathematical models: Poisson’s equation and
advection-diffusion equation, discretized through the Finite Difference Method (FDM). The numerical schemes
used were CDS and CDS-2 (Central Difference Scheme), for first and second order derivatives, respectively. The
systems of equations resulting from the discretizations were solved by the TriDiagonal Matrix Algorithm (TDMA)
and Gauss-Seidel (GS). The solver TDMA was used to obtain the exact solution and the solver GS to estimate the
errors at each iteration. Two variables were chosen to evaluate the results: function value at the central grid point
(local) and the average value of the function (global). The codes were implemented in Fortran 95, with quadruple
precision, in the Microsoft Visual Studio Community 2013. The results showed that the proposed estimator has a
performance similar to the ones already existing in the literature and that it is necessary to improve the estimates in
the initial ranges of the iterative process and in the final ranges, where the rounding error becomes more significant.