• Journal of applied mathematics & informatics
• /
• 제32권5_6호
• /
• pp.567-584
• /
• 2014

In this paper, we consider approximation of eigenelements of a two dimensional compact integral operator with a smooth kernel by discrete collocation and iterated discrete collocation methods. By choosing numerical quadrature appropriately, we obtain convergence rates for gap between the spectral subspaces, and also we obtain superconvergence rates for eigenvalues and iterated eigenvectors. We then apply Richardson extrapolation to obtain further improved error bounds for the eigenvalues. Numerical examples are presented to illustrate theoretical estimates.

• 대한수학회지
• /
• 제51권4호
• /
• pp.679-702
• /
• 2014

Investigation of the numerical solution of singularly perturbed turning point problems dates back to late 1970s. However, due to the presence of layers, not many high order schemes could be developed to solve such problems. On the other hand, one could think of applying the convergence acceleration technique to improve the performance of existing numerical methods. However, that itself posed some challenges. To this end, we design and analyze a novel fitted operator finite difference method (FOFDM) to solve this type of problems. Then we develop a fitted mesh finite difference method (FMFDM). Our detailed convergence analysis shows that this FMFDM is robust with respect to the singular perturbation parameter. Then we investigate the effect of Richardson extrapolation on both of these methods. We observe that, the accuracy is improved in both cases whereas the rate of convergence depends on the particular scheme being used.

• Structural Engineering and Mechanics
• /
• 제55권4호
• /
• pp.885-900
• /
• 2015

This paper introduces Romberg-Richardson's method as one of the numerical integration tools for computation of stress intensity factor in a pre-cracked specimen subjected to a complex stress field across the crack faces. Also, the computation of stress intensity factor for various stress fields using existing three methods: average stress over interval method, piecewise linear stress method, piecewise quadratic method are modified by using Richardson extrapolation method. The direct integration method is used as reference for constant and linear stress distribution across the crack faces while Gauss-Chebyshev method is used as reference for nonlinear distribution of stress across the crack faces in order to obtain the stress intensity factor. It is found that modified methods (average stress over intervals-Richardson method, piecewise linear stress-Richardson method, piecewise quadratic-Richardson method) yield more accurate results after a few numbers of iterations than those obtained using these methods in their original form. Romberg-Richardson's method is proven to be more efficient and accurate than Gauss-Chebyshev method for complex stress field.

• 대한수학회지
• /
• 제49권3호
• /
• pp.549-569
• /
• 2012

In this paper asymptotic error expansions for mixed finite element approximations to a class of second order elliptic optimal control problems are derived under rectangular meshes, and the Richardson extrapolation of two different schemes and interpolation defect correction can be applied to increase the accuracy of the approximations. As a by-product, we illustrate that all the approximations of higher accuracy can be used to form a class of a posteriori error estimators of the mixed finite element method for optimal control problems.

• Structural Engineering and Mechanics
• /
• 제16권2호
• /
• pp.219-240
• /
• 2003

One of the most versatile approaches for analyzing the dynamic behavior of structural systems is direct time integration of semi-discrete equations of motion. However responses computed by time integration are generally inexact and hence the corresponding errors would rather be studied in advance. In spite of the various error estimation formulations that exist in the literature, it is accepted practice to repeat the analyses with smaller time steps, followed by a comparison between the results. In this paper, after a review of this simple method and disregarding the round-off errors, a more efficient, reliable and yet simple method for estimating errors and enhancing the accuracy is proposed. The main objectives of this research are more realistic error estimation based on the concept of convergence, approximately controlling the reliability by comparing the actual rate of convergence with the integration method's order of accuracy, and enhancement of reliability by applying Richardson's extrapolation. Starting from the errors at specific time instants, the study is then generalized to cases in which the errors should be estimated and decreased at specific events e.g. peak responses. Numerical study illustrates the efficacy of the proposed method.

• 한국컴퓨터정보학회논문지
• /
• 제3권4호
• /
• pp.55-62
• /
• 1998

이 논문은 수치해석에서 적분값을 구하는데 이용되는 Romberg 적분법이 많은 계산량으로 인하여 소프트웨어적인 방법으로는 처리 속도가 떨어지므로 수치처리를 위한 툴 키트를 사용시 처리속도가 떨어진다. 그래서 이 논문에서는 시스토릭어레이를 이용하여 Romberg 적분법에 적분값을 구하는 새로운 하드웨어를 제안하였다. 이 새로운 하드웨어는Romberg 적분법이 2단계로 나누어져있어서 2단계의 시스토릭어레이로 설계를 하였다. 첫번째 단계는 사다리꼴 적분법에 의해서 근사치를 구하고, 두 번째는 단계는 구해진 적분값을수렴속도도 빠르고 근사 값을 정확하게 하기 위해서 오차의 위수를 높여 가는 방법에 많이사용하는 Richardson의 외삽법을 적용하여 적분값을 구하는 것이다.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• 제15권1호
• /
• pp.1-17
• /
• 2011

A new method for option pricing based on the trinomial tree method is introduced. The new method calculates the local average of option prices around a node at each time, instead of computing prices at each node of the trinomial tree. Local averaging has a smoothing effect to reduce oscillations of the tree method and to speed up the convergence. The option price and the hedging parameters are then obtained by the compact scheme and the Richardson extrapolation. Computational results for the valuation of European and American vanilla and barrier options show superiority of the proposed scheme to several existing tree methods.

• Industrial Engineering and Management Systems
• /
• 제10권2호
• /
• pp.170-177
• /
• 2011

We present an improved binomial method for pricing financial deriva-tives by using cell averages. After non-overlapping cells are introduced around each node in the binomial tree, the proposed method calculates cell averages of payoffs at expiry and then performs the backward valuation process. The price of the derivative and its hedging parameters such as Greeks on the valuation date are then computed using the compact scheme and Richardson extrapolation. The simulation results for European and American barrier options show that the pro-posed method gives much more accurate price and Greeks than other recent lattice methods with less computational effort.

• 한국전산유체공학회:학술대회논문집
• /
• 한국전산유체공학회 2008년도 학술대회
• /
• pp.79-83
• /
• 2008

An approach to CFD code validation is developed that gives proper consideration to experimental and simulation uncertainties. The comparison errors include the difference between the data, simulation values and represents the combination of all errors. The uncertainties of modeling and numerical analysis in the CFD prediction were estimated by a Coleman's theory. In this paper, the numerical solutions are calculated by A-type standard uncertainty and Richardson extrapolation Method.

• 한국전산유체공학회:학술대회논문집
• /
• 한국전산유체공학회 2008년 추계학술대회논문집
• /
• pp.79-83
• /
• 2008

An approach to CFD code validation is developed that gives proper consideration to experimental and simulation uncertainties. The comparison errors include the difference between the data, simulation values and represents the combination of all errors. The uncertainties of modeling and numerical analysis in the CFD prediction were estimated by a Coleman's theory. In this paper, the numerical solutions are calculated by A-type standard uncertainty and Richardson extrapolation Method.