Asymptotic upper bounds for the errors of Richardson extrapolation with practical application in approximate computations

Abstract

SUMMARYThe results produced by Richardson extrapolation, though, in general, very accurate, are inexact. Numerical evaluation of this inexactness and implementation of the evaluation in practice are the objectives of this paper. First, considering linear changes of errors in the convergence plots, asymptotic upper bounds are proposed for the errors. Then, the achievement is extended to the results produced by Richardson extrapolation, and finally, an error-controlling procedure is proposed and successfully implemented in approximate computations originated in science and engineering.

On Time History Analyis With Steps Larger Than the Steps of Earthquake Records Independent From the Frequency Contents

“…[51], the accuracy of time integration analysis is to be controlled after the analysis. The most conventional comment in this regard is repetition of the analysis with half steps and somehow comparing the responses [17,23]; for other practical comments, for instance, see [52,53]. In other words, traditionally, after the time integration analysis, we would rather check the accuracy by somehow comparing the responses with some other responses supposed to be more accurate.…”

More Versatility for an Integration-Step-Size-Enlargement Technique in Time Integration Analysis

“…Convergence is both the first essentiality in general numerical computation [17][18][19], and also the key basis in the recently proposed technique [16]. In numerical investigation of convergence and order of accuracy, for problems subjected to digitized excitations, e.g.…”

“…equations of motion subjected to digitized earthquake records (Eqs. (1)), it is conventional to implement linear interpolation of the digitized records in analyses with smaller steps [12,[19][20][21]. In view of this idea, we can convert an earthquake record digitized at steps equal to t  , to a record digitized at smaller steps, by linear interpolation, and expect no loss of accuracy in time integration analysis (compared to the exact responses); though in the price of more computational cost.…”

A Different Look at the Richardson Extrapolation Leading to a New Proposition

“…increase the order of accuracy and rate of convergence [3,4]. This is a significant achievement addressed in the literature as the most usefulness of Richardson extrapolation for practical computations or turning lead to gold [5,6].…”

“…Some of the applications of Richardson extrapolation are different advanced integration methods, e.g. the Romberg method, and a recent error estimation approach [4,6]. Order of accuracy and its increase would lose its meaning without convergence.…”