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Topic: Richardson extrapolation

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	Examining Spatial (Grid) Convergence
		Richardson extrapolation is a method for obtaining a higher-order estimate of the continuum value (value at zero grid spacing) from a series of lower-order discrete values.
		Richardson extrapolation can be applied for the solution at each grid point, or to solution functionals, such as pressure recovery or drag.
		Richardson extrapolation is based on a Taylor series representation as indicated in Eqn.
	www.grc.nasa.gov /WWW/wind/valid/tutorial/spatconv.html (2858 words)

	Re: Richardson extrapolation
		Richardson extrapolation - pc, Fri, 2 Jun 2006, 12:03 p.m.
		Re: Richardson extrapolation - ganesh, Sat, 3 Jun 2006, 6:19 a.m.
		Re: Richardson extrapolation - ganesh, Tue, 6 Jun 2006, 12:40 p.m.
	www.cfd-online.com /Forum/main_archive.cgi?read=45815 (269 words)

	Richardson Extrapolation
		Inthe finite difference method, a Richardson extrapolation can be used to improve the accuracy.
		Clearly for small ∆x the plot becomes linear; this suggests that the ∆x is small enough that the theory applies, and the missing terms represented by.
		The extrapolated finite difference solution gives the same answer, which verifies that we have solved the correct problem.
	faculty.washington.edu /finlayso/ebook/bvp/bvp_fd/errors/Richardson.htm (982 words)

	Lesson 19
		Richardson Extrapolation and Romberg Integration # ------------------------------------------------------------ # # Last time we defined the Newton-Cotes rules for numerical # integration of int(f(x), x=a..b).
		But after this the Richardson Extrapolation results # are not Newton-Cotes rules.
		It appears that the # error estimates in the third row are not trustworthy (and # indeed Ee[4,3] is quite bad).
	www.ugrad.math.ubc.ca /coursedoc/m210/lesson19.html (839 words)

	Isospectral Drums (Site not responding. Last check: 2007-10-12)
		To perform the Richardson extrapolation you need three meshes of different levels of refinement.
		Notice that the extrapolated values are all smaller than the corresponding non-extrapolated values.
		Finally, compare the Richardson extrapolation results for the first six eigenvalues of the two domains and note that they agree to at least five decimal places.
	www-math.cudenver.edu /~jmandel/doc/modlib/modlib61.htm (880 words)

	AMTH247 Lecture 18 Numerical Differentiation Richardson Extrapolation
		The extrapolated result shows the features of the second order methods in the previous example; they reach maximum accuracy more quickly and have higher maximum accuracy.
		Romberg integration involves the systematic application of Richardson Extrapolation to the composite trapezoidal rule.
		The first column of this array are the initial compound trapezoidal approximations, the second column the first extrapolation etc. Each approximation in the array is computed from the approximations immediately to its left and upper left.
	turing.une.edu.au /~amth247/Lectures_2003/Lecture_18/lecture (751 words)

	Abstract (Site not responding. Last check: 2007-10-12)
		The technique of Richardson extrapolation, which has previously been used on time-independent problems, is extended so that it can also be used on time-dependent problems.
		The technique presented is completed in the sense that the extrapolated solution is calculated at all spatial grid nodes which coincide with the nodes of the finest grid considered.
		The examples show that extrapolation can be an easy and efficient way in which to produce accurate numerical solutions to time-dependent problems.
	www.nceas.ucsb.edu /~richards/R1997Abstract.html (98 words)

	Extrapolcation & Errors
		Richardson extrapolation can be used to improve the accuracy of a method.
		Using the extrapolated answer gives an actual error of only 0.0004, whereas the solution with smallest time step had an error of 0.00939, or almost 23 times larger.
		We can also plot the result versus t and, if t is small enough for the asymptotic results to be valid, we should get a straight line.
	faculty.washington.edu /finlayso/ebook/ivp/explicit/error.htm (429 words)

	Practical Extrapolation Methods - Cambridge University Press
		Thus, to approximate limits with reasonable accuracy, it is necessary to compute a large number of terms, and this is in general costly.
		These limits can be approximated economically and with high accuracy by applying suitable extrapolation (or convergence acceleration) methods to a small number of terms.
		This state-of-the art reference for mathematicians, scientists and engineers is concerned with the coherent treatment, including derivation, analysis, and applications, of the most useful scalar extrapolation methods.
	www.cambridge.org /catalogue/catalogue.asp?isbn=0521661595 (409 words)

	Richardson extrapolation
		Does anyone have experience with applying Richardson extrapolation to compute estimated errors for the whole flowfield?
		I'm not referring to globally integrated quantities such as lift/drag, put pointwise application of Richardson extrapolation at each (x,y,z) on the finest grid.
		One must have solution data at the same (x,y,z) from each of the meshes used, so some interpolation is required.
	www.cfd-online.com /Forum/main_archive.cgi?read=45786 (213 words)

	MGNet Preprints (Site not responding. Last check: 2007-10-12)
		Extrapolation is a well-known numerical technique for raising the approximation order.
		Several variants of extrapolation can be used for elliptic partial differential equations.
		The basic algorithmic variants are Richardson extrapolation, truncation error extrapolation and extrapolation of the functional.
	www.mgnet.org /mgnet/papers/Ruede/extra.html (104 words)

	Scientific WorkPlace as a Tool to Evaluate Definite Integral Using Trapezoidal Rule
		In general, of course, we do not know how good an approximation this value is, and the usual practice is to continue interval-halving until two successive extrapolated values agree to the accuracy required.
		We can also look at the rate at which the computational error is reducing in Richardson extrapolation.
		So, when using Richardson extrapolation, the computational error is reduced to about 6% each time when the number of intervals is doubled.
	archives.math.utk.edu /ICTCM/EP-9/C5/html/paper.html (1368 words)

	[No title]
		Computational aspects of Richardson's extrapolation methods implementation and nonlinear iterations are investigated.
		Section 5 describes the multigrid algorithms with Richardson's extrapolation that allow us to obtain the solution of higher-order accuracy on the basis of simple piecewise linear approximations.
		The results of applying the Richardson's extrapolation algorithms are shown below.
	www.mgnet.org /mgnet/conferences/ParMGM98/Papers/trofimov.doc (1685 words)

	Home Page of Edward J. Allen
		William E. Abbott, M.S., Thesis: Richardson extrapolation of a positive method for numerically solving the transport equation in spherical geometry, May, 1989.
		D., Dissertation: Richardson extrapolation of difference methods in financial option valuation and the development and analysis of shooting methods for Stratonovich stochastic differential equation systems with boundary conditions, August 2003.
		Drew, M.S., Thesis: An investigation of climatic and geographic factors on the growth and spread of Chytrid fungus on amphibian populations in Australia, co-advisor: L. Allen, December, 2004.
	www.math.ttu.edu /~eallen (2123 words)

	Romberg Integration (Site not responding. Last check: 2007-10-12)
		This module illustrates Romberg integration, which is a numerical quadrature technique based on repeated Richardson extrapolation of values obtained using the composite trapezoid rule with successively halved mesh spacing.
		The previously computed values used in the extrapolation are indicated in blue and magenta, and the resulting extrapolated value in red, both in the graph and the triangular array.
		The correct numerical value I of the integral is shown below the selected function for comparison with the values in the triangular array, and is also plotted as a fl point (which may be obscured by the red point) in the right panel.
	www.cse.uiuc.edu /eot/modules/integration/romberg (281 words)

	Richardson's Extrapolation
		The central difference formula for the first derivative, based on five points is a linear combination of
		Richardson's extrapolation relates the five point rule and the three point rule,
		Various Scenarios and Animations for Richardson's Extrapolation and higher derivatives.
	math.fullerton.edu /mathews/n2003/RichardsonExtrapMod.html (271 words)

	Different methods for solving ODES (Site not responding. Last check: 2007-10-12)
		These methods propagate a solution over an interval by combining information from several small steps and then combining the information to match a higher order taylor expansion.
		Richardson extrapolation methods take a computed result and expolate them to what they would have been had the step size been much smaller.
		Predictor corrector methods store the solutions, extrapolate the next solution and then correct the extrapolation using derivative information at the new point.
	web.mit.edu /course/other/cdsdev/html/Ode/types.html (82 words)

	File PROB_ND.F
		A common use of special error testing is to mark contact discontinuities (locations of high vorticity) as requiring refinement.
		If this parameter is set greater than or equal to zero, then both Richardson and special error testing are performed.
		the difference between the two representations is used for richardson error extrapolation.
	www.c3.lanl.gov /~pernice/hamr/docs/node11.html (1085 words)

	Worksheet 24
		# # # WORKSHEET # 24 # # RICHARDSON EXTRAPOLATION # # -------------------------------------------------------------------------------- # In this worksheet we describe a quite elementary method, called Romberg integration, for # improving on the accuracy of numerical methods of integration.
		We apply it first to the # trapezoidal rule.
		Recall that n must be even for Simpson's rule.
	www.ugrad.math.ubc.ca /coursedoc/m210/worksh24.html (1085 words)

	MCS 471 Exam 2(a) (Site not responding. Last check: 2007-10-12)
		Use central differences and Richardson extrapolation to compute a sixth order approximation for the derivative of arctan(x) at x = 0.4.
		Start with h=0.1 and divide h by 2 (i.e.: r = 1/2) to build the first column of the extrapolation table.
		How many decimal places are correct in your final answer from Richardson extrapolation?
	www.math.uic.edu /~jan/MCS471/Lec33/exam2a.html (267 words)

	EE31 Matlab Examples (Site not responding. Last check: 2007-10-12)
		Some of them are already experienced in the course notes: midpoint rule, Euler-Maclaurin, trapezoidal and Simpson rule.
		In this section we consider again in a more broader way the trapezoidal rule, the Richardson extrapolation, Simpson’s rules and many other examples not covered in the course.
		With a midpnt function like the one given in the course notes (midpnt.m) the Richardson extrapolation can be executed like in midpnt_richard.m and midpnt_richard.m (see class notes).
	www.eeng.dcu.ie /~ee317/Matlab_Examples/integr/tutinfo[1].htm (1222 words)

	Richardson Extrapolation
		Richardson Extrapolation is an idea which can often be used to improve the results of a numerical method: from a method of order O(h
		The idea of Richardson extrapolation is to take two different values of h, and eliminate the A term.
		Since p = 1, the Richardson approximation would be
	www.math.ubc.ca /~israel/m215/rich/rich.html (236 words)

	On the efficient application of the repeated Richardson extrapolation technique to option pricing
		On the efficient application of the repeated Richardson extrapolation technique to option pricing
		Richardson extrapolation (RE) is a commonly used technique in financial applications for accelerating the convergence of numerical methods.
		Particularly in option pricing, it is possible to refine the results of several approaches by applying RE, in order to avoid the difficulties of employing slowly converging schemes.
	ideas.repec.org /p/vnm/wpaper/147.html (528 words)

	High-Order Accurate Decomposition Methods Based on Richardson's Extrapolation for a Singularly Perturbed Elliptic ... (Site not responding. Last check: 2007-10-12)
		High-Order Accurate Decomposition Methods Based on Richardson's Extrapolation for a Singularly Perturbed Elliptic Reaction-Diffusion Equation on a Strip
		We consider the Dirichlet problem on a strip for a singularly perturbed elliptic equation of reaction-diffusion type.For such a problem well-known finite difference schemes converge $\eps$-uniformly with the order of accuracy not higher than second.
		For the above Richardson scheme we construct a decomposition method on overlapping subdomains for which $\eps$-uniform accuracy of the Richardson scheme is preserved.
	www.maths.tcd.ie /report_series/abstracts/tcdm0204.html (107 words)

	Remarks (Site not responding. Last check: 2007-10-12)
		We can predict that given a function with two variables, there should be an optimal choice for picking the uniform regular matrices, a
		We have the Richardson Extrapolation for the quadrature in one dimension as we mentioned earlier.
		We should be able to apply the Richardson Extrapolation for the two dimensional quadrature to speed up the convergence.
	www.runet.edu /~wyang/mypaper/node22.html (67 words)

	Table of contents for Library of Congress control number 2002024669
		Table of contents for Practical extrapolation methods : their mathematical theory and application / Avram Sidi.
		Bibliographic record and links to related information available from the Library of Congress catalog
		Library of Congress subject headings for this publication: Extrapolation
	www.loc.gov /catdir/toc/cam024/2002024669.html (215 words)

	The Forth Scientific Library (Site not responding. Last check: 2007-10-12)
		Polynomial and Rational function interpolation and extrapolation, V1.2.
		Adaptive integration using trapezoidal rule with Richardson extrapolation
		Adaptive integration using 3 point Gauss-Legendre with Richardson extrapolation
	www.taygeta.com /fsl/scilib.html (446 words)

	AMTH247 Lecture 21 Differential Equations III
		Richardson extrapolation can be applied to the numerical solution of initial value problems.
		The following is a Scilab script implementing Richardson extrapolation applied to Euler's method.
		We need to careful about the size of vectors and use expressions like
	turing.une.edu.au /~amth247/Lectures_2003/Lecture_21/lecture (602 words)

	Verification and Validation in Computational Science and Engineering
		6.27 Other Examples of Careful Use of Richardson Extrapolation
		Appendix D. CFD Parody: Will the Wind Tunnel Replace the Computer?
		Appendix E. A Biographical Sketch of Lewis Fry Richardson
	kumo.swcp.com /hermosa/html/vvcse.html (734 words)

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