Richardson extrapolation

Richardson extrapolation can be used to improve the accuracy of a method. Suppose we step forward one step At with a pth-order method. Then redo the problem, this time stepping foi ward from the same initial point, but in two steps of length Af/2, thus ending at the same point. Call the solution of the one-step calculation y and the solution of the two-step calculation yo. Then an improved solution at the new time is given by... 

Figure 8.19 F.llingham diagram for the free energy of formation of metallic oxides. (After F. D. Richardson and J. H. F. Jeffes, J. Iron Steel Inst. 160, 261 (1948).) The oxygen dissociation pressure of a given M - MO system at a given temperature is obtained by joining on the lop left hand to the appropriate point on the M-MO frec-energy line, and extrapolating to the scale on the right hand ordinate for POi (atm).

If it is known that a particular form of relation, such as the power-law model, is applicable, it is not necessary to maintain a constant shear rate. Thus, for instance, a capillary tube viscometer can be used for determination of the values of the two parameters in the model. In this case it is usually possible to allow for the effects of wall-slip by making measurements with tubes covering a range of bores and extrapolating the results to a tube of infinite diameter. Details of the method are given by Farooqi and Richardson. 21 ... 

Using the solutions with 40, 20 and 10 grid points for Richardson extrapolation with Equation (A. 16) we have the following system of equations in terms of extrapolated value and extrapolation constants of exit mass fraction of hydrogen. 

Extrapolation is an old technique invented by Richardson in 1927 [469]. Generally it makes use of known error orders to increase accuracy. In the present context, its application is based on the first-order method BI, mentioned above. One defines a notation in terms of operations L on the variable y(t), the operation being that of taking a step forward in time. Thus, the notation L y t) or, in terms of discrete time steps where one whole interval is St, Liyn, means a single step of one interval (the 1 being indicated by the subscript on L). The simplest variant is then the application of operation L, followed by two operations, t jA, that is, two consecutive steps of half St (again starting the first from yn), and finally a linear combination of the two results ... 

The procedure can be iterated to calculate the next terms in the C/i series and so on and, therefore, to further improve the solution. By doing so, the Richardson extrapolation formulae are obtained and they are referred to as the Romberg method in the special case of definite integrals. 

The values of T and the corresponding h can be considered as support points of a polynomial interpolation. From this perspective, the Richardson extrapolation corresponds to the polynomial prediction for h = 0. 

The algorithm usually adopted to prepare the points for the extrapolation, obtained by varying h, is a variant of the central point method proposed by Gragg (Stoer and BuUrsch, 1983), while the extrapolation for -> 0 is performed either with the Richardson method applied to polynomials or the Bulirsch-Stoer (Stoer and BuUrsch, 1983) method applied to rational functions. 

Once the value of is calculated for different Hn, an extrapolation for A = 0 is performed. For the extrapolation, we can use a polynomial approximation with the Neville algorithm, which is equivalent to the Richardson extrapolation for this specific case, or better still, a rational function using the Bulirsch-Stoer algorithm. The convergence of the method can be assessed by comparing the two different values of the extrapolation. 

In principle, we could determine h and cr immediately from the slope and y intercept of the linear part of Fig. 1. In practice, one must use more sophisticated procedures in order to extrapolate the ratios to their asymptotic limit, since the actual Ri do not become precisely linear until i is very large. We have available [9] the expansion coefficients for through order 49, although the higher-order values suffer some from roundoff error. Using Neville-Richardson extrapolation, we have been able to establish [9,10] that a = 1/2, so that (j) represents a square-root branch point. 

Equation 16 forms the basis of the Richardson extrapolation formula [6] commonly used in numerical analysis, for improving the accuracy of the numerical solutions by accommodating for the truncation error estimates. 

The enzymatic steps required for DNA synthesis in vitro have been carried to the point where one can make extrapolations to the mechanism of in vivo synthesis (see Kornberg, 1969 Richardson, 1969). In order to synthesize DNA, a cell must produce the necessary deoxyribonucleoside triphosphates. The requirement for these precursors for DNA synthesis makes their production a possibly important part of the control of DNA synthesis (see Lark, 1963 Larsson and Reichard, 1967). Cells exhibit control over both... 

The Richardson extrapolation method is a useful tool to obtain estimates of a continuous high-order value through a series of discrete values of lower order. A certain amount/ can be expressed in series as... 

Finite-difference methods operating on a grid consisting of equidistant points ( Xi, Xi = ih + Xq) are known to be one of the most accurate techniques available [496]. Additionally, on an equidistant grid all discretized operators appear in a simple form. The uniform step size h allows us to use the Richardson extrapolation method [494,497] for the control of the numerical truncation error. Many methods are available for the discretization of differential equations on equidistant grids and for the integration (quadrature) of functions needed for the calculation of expectation values. 

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