Design of second order digital differentiator using Richardson extrapolation and fractional delay

2008 ◽  
Author(s):  
Chien-Cheng Tseng ◽  
Su-Ling Lee
Keyword(s):  
Richardson Extrapolation ◽  
Second Order ◽  
Digital Differentiator ◽  
Fractional Delay

scholarly journals Study on Accuracy of the High-Resolution Schemes

2014 ◽  
Vol 6 ◽  
pp. 905053
Author(s):  
Yawen Tang ◽  
Bo Yu ◽  
Jianyu Xie ◽  
Jingfa Li ◽  
Peng Wang
Keyword(s):  
High Resolution ◽  
Numerical Solution ◽  
Richardson Extrapolation ◽  
Second Order ◽  
Order Accuracy ◽  
Order Of Accuracy ◽  
Numerical Examples ◽  
High Resolution Schemes ◽  
Artificial Diffusion ◽  
Convection Dominated

The high-resolution (HR) schemes have been widely used as they can achieve the numerical solution without oscillation and artificial diffusion, especially for convection-dominated problems. However, there still have arguments about the order of accuracy of HR schemes, especially about the extreme value of the solution. In this paper, it is proved that any HR scheme designed in the NVD diagram has second-order accuracy when its combined segments totally locate in the BAIR region. In other words, it has been verified in our study that the segments, which have low-order accuracy when independently employed, have at least second-order accuracy when locate in BAIR region by analysis of two implementation methods of HR scheme and also a number of numerical examples. Meanwhile Richardson extrapolation has been used to estimate the order of accuracy of HR schemes which achieve the same conclusion.

WHAT A DIFFERENCE ONE PROBABILITY MAKES IN THE CONVERGENCE OF BINOMIAL TREES

2020 ◽  
Vol 23 (06) ◽  
pp. 2050040
Author(s):  
GUILLAUME LEDUC ◽  
KENNETH PALMER
Keyword(s):  
Richardson Extrapolation ◽  
Second Order ◽  
Barrier Options ◽  
Numerical Evidence ◽  
First Order ◽  
Black Scholes ◽  
Binomial Trees ◽  
Smooth Convergence ◽  
First Time ◽  
Flexible Model

In the [Formula: see text]-period Cox, Ross, and Rubinstein (CRR) model, we achieve smooth convergence of European vanilla options to their Black–Scholes limits simply by altering the probability at one node, in fact, at the preterminal node between the closest neighbors of the strike in the terminal layer. For barrier options, we do even better, obtaining order [Formula: see text] convergence by altering the probability just at the node nearest the barrier, but only the first time it is hit. First-order smooth convergence for vanilla options was already achieved in Tian’s flexible model but here we show how second order smooth convergence can be achieved by changing one probability, leading to convergence of order [Formula: see text] with Richardson extrapolation. We illustrate our results with examples and provide numerical evidence of our results.

scholarly journals Fourth-order stable central difference with Richardson extrapolation method for second-order self-adjoint singularly perturbed boundary value problems

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Muslima Kedir Siraj ◽  
Gemechis File Duressa ◽  
Tesfaye Aga Bullo
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Mesh Size ◽  
Perturbation Parameter ◽  
Richardson Extrapolation ◽  
Second Order ◽  
Singularly Perturbed ◽  
Algebraic Equations ◽  
Central Difference ◽  
Convergence Of The Scheme

AbstractThis study introduces a stable central difference method for solving second-order self-adjoint singularly perturbed boundary value problems. First, the solution domain is discretized. Then, the derivatives in the given boundary value problem are replaced by finite difference approximations and the numerical scheme that provides algebraic systems of equations is developed. The obtained system of algebraic equations is solved by Thomas algorithm. The consistency and stability that guarantee the convergence of the scheme are investigated. The established convergence of the scheme is further accelerated by applying the Richardson extrapolation which yields sixth order convergent. To validate the applicability of the method, two model examples are solved for different values of perturbation parameter ε and different mesh size h. The proposed method approximates the exact solution very well. Moreover, the present method is convergent and gives more accurate results than some existing numerical methods reported in the literature.

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