China
Africa
Explore chapters and articles related to this topic
Numerical Integration
Published in Santanu Saha Ray, Numerical Analysis with Algorithms and Programming, 2018
It may be noted that I2 is more accurate than I1 if h2 < h1 and therefore, I12 is an improved approximation over I2. Since α > 1, we see that I12 > I2 if I1 < I2 and I12 < I2 if I2 < I1. Thus, in either case, I12 lies outside interval [I1,I2] or [I2, I1] as the case may be, that is, I12 is computed from I1 and I2 with the help of extrapolation. This process is known as Richardson extrapolation.
Electromagnetics
Published in Arun G. Phadke, Handbook of Electrical Engineering Calculations, 2018
Another technique that increases accuracy for some sequences and which is used in Romberg’s Method for numerical quadrature (integration), is Richardson extrapolation [37, 38]. Richardson extrapolation is based on the idea that the error in a numerical process which depends on a sampling interval h is proportional to h raised to an integer power. Suppose for example that () f(h)=ftrue+E(h)
Numerical Differentiation and Integration
Published in Azmy S. Ackleh, Edward James Allen, Ralph Baker Kearfott, Padmanabhan Seshaiyer, Classical and Modern Numerical Analysis, 2009
Azmy S. Ackleh, Edward James Allen, Ralph Baker Kearfott, Padmanabhan Seshaiyer
The above process is called Richardson extrapolation, and can be performed whenever the error expansion has the form (6.39) or (6.41). Richardson extrapolation is useful in numerical integration, numerical solution of integral equations and initial-value problems, numerical methods for solving partial differential equations, and stochastic differential equations.
Almost second-order uniformly convergent numerical method for singularly perturbed convection–diffusion–reaction equations with delay
Published in Applicable Analysis, 2023
Mesfin Mekuria Woldaregay, Gemechis File Duressa
We apply the Richardson extrapolation technique to accelerate the rate of convergence of the scheme in spatial direction. Richardson Extrapolation is a convergence acceleration technique which involves the combination of two computed approximations of solution. From (37), we have where and are exact and approximate solutions, respectively. Let denote an approximate solution on number of mesh points in the spatial direction by including the mid-points . From (38), we have So, this works for any gives where the remainders and are the order of . Combining (39) and (40) leads to which gives that is also an approximate solution. The error of the spatial discretization approximation in (41) becomes
Potential of statistical model verification, validation and uncertainty quantification in automotive vehicle dynamics simulations: a review
Published in Vehicle System Dynamics, 2022
Benedikt Danquah, Stefan Riedmaier, Markus Lienkamp
Numerical uncertainties arise in computer simulation because a computer discretises the continuous space and has a finite precision. Model verification deals with identifying the numerical uncertainties due to rounding errors, discretisation errors, etc. There are many verification techniques available in the literature. Especially in CFD simulations, the numerical uncertainties and the discretisation errors have a significant influence on the results. The Richardson extrapolation is a technique to quantify the discretisation error and convert it to an uncertainty [54]. In vehicle-dynamics simulations the numerical influences are often negligible [16,p.14].
Verification of numerical solutions of thermal radiation problems in participating and nonparticipating media
Published in Numerical Heat Transfer, Part B: Fundamentals, 2023
Antonio Carlos Foltran, Carlos Henrique Marchi, Luís Mauro Moura
When appropriated, an extrapolation method produces, from a certain sequence, a new sequence that converges to the limit of the first one, but with higher order [9]. The Richardson extrapolation can be used as a post-processing technique, increasing the accuracy of results even if low order formulas are employed. It can reduce the discretization error even if the numerical results were obtained in relatively coarse grids, requiring only that these results are already in the monotonic convergence region.