Limitations of Richardson Extrapolation and Some Possible Remedies

2005 ◽  
Vol 127 (4) ◽  
pp. 795-805 ◽  
Author(s):  
Ismail Celik ◽  
Jun Li ◽  
Gusheng Hu ◽  
Christian Shaffer
Keyword(s):  
Error Function ◽  
Finite Difference Methods ◽  
Taylor Series Expansion ◽  
Variable Coefficients ◽  
Oscillatory Behavior ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Minimum Number ◽  
Manufactured Solution ◽  
New Perspective

The origin of oscillatory convergence in finite difference methods is investigated. Fairly simple implicit schemes are used to solve the steady one-dimensional convection-diffusion equation with variable coefficients, and possible scenarios are shown that exhibit the oscillatory convergence. Also, a manufactured solution to difference equations is formulated that exhibits desired oscillatory behavior in grid convergence, with a varying formal order of accuracy. This model-error equation is used to statistically assess the performance of several methods of extrapolation. Alternative extrapolation schemes, such as the deferred extrapolation to limit technique, to calculate the coefficients in the Taylor series expansion of the error function are also considered. A new method is proposed that is based on the extrapolation of approximate error, and is shown to be a viable alternative to other methods. This paper elucidates the problem of oscillatory convergence, and brings a new perspective to the problem of estimating discretization error by optimizing the information from a minimum number of calculations.

Limitations of Richardson Extrapolation and Possible Remedies for Estimation of Discretization Error

2004 ◽  
Author(s):  
I. B. Celik ◽  
J. Li ◽  
G. Hu ◽  
C. Shaffer
Keyword(s):  
Error Function ◽  
Finite Difference Methods ◽  
Taylor Series Expansion ◽  
Variable Coefficients ◽  
Oscillatory Behavior ◽  
Discretization Error ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Minimum Number ◽  
Manufactured Solution

The origin of oscillatory convergence in finite difference methods is investigated. Fairly simple implicit schemes are used to solve the steady convection diffusion equation with variable coefficients and possible scenarios are shown that exhibit the oscillatory convergence. Also a manufactured solution to difference equations is formulated that exhibit desired oscillatory behavior in gird convergence with varying formal order of accuracy. This model-error equation is used to assess statistically the performance of several methods of extrapolation. Alternative extrapolation schemes such as the deferred extrapolation to limit technique to calculate the coefficients in the Taylor series expansion of the error function are also considered. A new method is proposed that is based on the extrapolation of approximate error and shown to be a viable alternative to the other methods. This paper elucidates the problem of oscillatory convergence and brings a new look into the problem of estimating discretization error by optimizing the information from a minimum number of calculations.

scholarly journals Novel finite point approach for solving time-fractional convection-dominated diffusion equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Xiaomin Liu ◽  
Muhammad Abbas ◽  
Honghong Yang ◽  
Xinqiang Qin ◽  
Tahir Nazir
Keyword(s):  
Variable Coefficients ◽  
Numerical Dispersion ◽  
Discrete Equation ◽  
Finite Point ◽  
Spatial Direction ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Tailored Finite Point Method ◽  
L1 Approximation ◽  
The Stability

AbstractIn this paper, a stabilized numerical method with high accuracy is proposed to solve time-fractional singularly perturbed convection-diffusion equation with variable coefficients. The tailored finite point method (TFPM) is adopted to discrete equation in the spatial direction, while the time direction is discreted by the G-L approximation and the L1 approximation. It can effectively eliminate non-physical oscillation or excessive numerical dispersion caused by convection dominant. The stability of the scheme is verified by theoretical analysis. Finally, one-dimensional and two-dimensional numerical examples are presented to verify the efficiency of the method.

Wavelet method for a class of fractional convection-diffusion equation with variable coefficients

2010 ◽  
Vol 1 (3) ◽  
pp. 146-149 ◽  
Author(s):  
Yiming Chen ◽  
Yongbing Wu ◽  
Yuhuan Cui ◽  
Zhuangzhuang Wang ◽  
Dongmei Jin
Keyword(s):  
Diffusion Equation ◽  
Variable Coefficients ◽  
Wavelet Method ◽  
Convection Diffusion ◽  
Convection Diffusion Equation

scholarly journals ERROR ANALYSIS IN THE NUMERICAL SOLUTION OF 3D CONVECTION-DIFFUSION EQUATION BY FINITE DIFFERENCE METHODS

2009 ◽  
Vol 8 (1) ◽  
pp. 12 ◽  
Author(s):  
E. C. Romão ◽  
J. B. Campos-Silva ◽  
L. F. M. De Moura
Keyword(s):  
Finite Difference ◽  
Error Analysis ◽  
Numerical Solution ◽  
Diffusion Coefficients ◽  
Finite Difference Methods ◽  
Reynolds Numbers ◽  
Central Difference ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Difference Methods

In this work an error analysis for numerical solution of 3D convectiondiffusionequation by finite difference methods has been done. The backward, the forward and the central difference schemes are applied for three applications: a case with diffusion dominant corresponding to high diffusion coefficients and two cases with convection dominant or with low diffusion coefficients. In the second application the convective coefficients are function only of the diffusion coefficient that in dimensionless form is named Reynolds numbers. In the third application the convective coefficients are function of both the Reynolds number and of the space. The three applications have analytical solutions to facilitate numerical comparisons of the solutions.

scholarly journals A New Way to Generate an Exponential Finite Difference Scheme for 2D Convection-Diffusion Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Caihua Wang
Keyword(s):  
Difference Scheme ◽  
Taylor Series Expansion ◽  
Convergent Series ◽  
Positive Type ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Extrapolation Algorithm ◽  
Convection Dominated ◽  
Insight Into ◽  
Point Stencil

The idea of direction changing and order reducing is proposed to generate an exponential difference scheme over a five-point stencil for solving two-dimensional (2D) convection-diffusion equation with source term. During the derivation process, the higher order derivatives alongy-direction are removed to the derivatives alongx-direction iteratively using information given by the original differential equation (similarly fromx-direction toy-direction) and then instead of keeping finite terms in the Taylor series expansion, infinite terms which constitute convergent series are kept on deriving the exponential coefficients of the scheme. From the construction process one may gain more insight into the relations among the stencil coefficients. The scheme is of positive type so it is unconditionally stable and the convergence rate is proved to be of second-order. Fourth-order accuracy can be obtained by applying Richardson extrapolation algorithm. Numerical results show that the scheme is accurate, stable, and especially suitable for convection-dominated problems with different kinds of boundary layers including elliptic and parabolic ones. The idea of the method can be applied to a wide variety of differential equations.

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