Order of accuracy

Some of the various methods of weighing the material in a railroad car, arranged approximately in increasing order of accuracy, are ... 

Do calibrate working standards against calibration standards that have at least an order of accuracy greater than the working standards. 

We have employed the Bragg-Williams approximation (BWA) to obtain rough estimates of the ordering/segregation critical temperatures. It is well known that the BWA usually overestimates critical temperatures (approximately by 20 %) in comparison with the exact value obtained from Monte Carlo simulations, or by other highly accurate methods of statistical mechanics. This order of accuracy Is nevertheless sufficient for our present purposes. 

Voltage transformers are classified into types AL, A, B, C and D, in descending order of accuracy. The ratio errors for small voltage changes (within ( 10 per cent of the rating) vary between 0.25 per cent and 5 per cent and the phase errors between ( 10 minutes and ( 60 minutes. 

Ratio and phase angle errors also occur due to the need for a portion of the primary current to magnetize the core and the requirement for a finite voltage to drive the current through the burden. These errors must be small. Current transformers are classified (in descending order of accuracy) into types AT, AM, BM, C and D. Ratio errors in class AT must be within the limits of ( 0.1 per cent for AT and ( 5 per cent for D, while the phase error limit on class A1 is ( 5 minutes to ( 2 minutes in types CM and C. 

With these relations established, we conclude that if the scheme is stable and approximates the original problem, then it is convergent. In other words, convergence follows from approximation and stability and the order of accuracy and the rate of convergence are connected with the order of approximation. 

Everything just said means that in establishing convergence and in determining the order of accuracy of a scheme it is necessary to evaluate the error of approximation, discover stability and then derive estimates of the form (22) known as a priori estimates. 

Let us stress once again that in order to estimate the order of accuracy of a scheme, it is necessary to estimate its order of accuracy only on a solution of the original problem. 

The method of test functions is quite applicable in verifying convergence and determining the order of accuracy and is stipulated by a proper choice of the function I7(x). Such a function is free to be chosen in any convenient way so as to provide the validity of the continuity conditions at every discontinuity point of coefficients. By inserting it in equation (1) of Section 1 we are led to the right-hand side / = kU ) — qU and the boundary values jj, — U(0) and = U 1). The solution of such a problem relies on scheme (4) of Section 1 and then the difference solution will be compared with a known function U x) on various grids. 

In order to evaluate the order of accuracy for scheme (2), it is necessary to make the accurate account of the error z = y u being viewed as a solution of problem (3). Moreover, the desirable estimate should be expressed in terms of the right-hand side ip. In this direction the error of approximation to ip x) is considered first. If k x) and q x),f x) then... 

The order of accuracy on non-equidistant grids. As before, the inner products are defined by... 

Adaptive grids may be of assistance in raising the order of accuracy without increasing the total number of nodes. If the subsidiary information... 

The co-equivalence property of homogeneous schemes lies in the main idea behind a new approach to the further estimation of the order of accuracy of a scheme on account of (9) or (10) its coefficients a, d, ip should be compared with coefficients d, d, (p of a simple specimen scheme, the accuracy order of which is well-known (see Section 7). 

The uniform convergence and the order of accuracy of a difference scheme. In the study of convergence and accuracy of scheme (2) we begin by placing the problem for the error... 

It is worth noting here that on the square grid (h = h. = h) this condition is automatically fulfilled. A proper choice of (p guarantees the sixth order of accuracy of scheme (9) on any such grid. Convergence of scheme (9) with the fourth order in the space C can be established without concern of condition (11). An alternative way of covering this is to construct an a priori estimate for A z p and then apply the embedding theorem (see Section 4). 

If scheme (II) is stable with respect to the right-hand side and approximates problem (I), then it converges and the order of accuracy coincides with the order of approximation. 

The order of accuracy of scheme (7) can be most readily evaluated with the aid of the representation for the error z = y u as... 

Because of the enormous range of difference approximations to an equation having similar asymptotic properties with respect to a grid step (the same order of accuracy or the number of necessary operations), their numerical realizations resulted in the appearance of different schemes for solving the basic problems in mathematical physics. 

If the results of such calculations are to be chemical value they must be sufficiently accurate. We know from both theory and experiment the kind of accuracy required if rates are to be estimated to a factor of ten, activation enrgies must be accurate to 1 kcal/mole. Since the methods available to us are crude, they must be tested empirically to see if they achieve this order of accuracy before any reliance can be placed in results obtained from them. 

We now should raise the question how does the accuracy order of a scheme depend on the approximation order on a solution Because the error zh = yh — uh solves problem (38) with the right-hand side tph (and vh), the link between the order of accuracy and the order of approximation is stipulated by the character of dependence of the difference problem solution upon the right-hand side. Let zh depend on tph and vh continuously and uniformly in h. In other words, if a scheme is stable, its order of accuracy coincides with the order of approximation. 

Theorem 1 If scheme (21) is correct and generates an approximation on an element u BA then it converges. More precisely, a solution yh of problem (21) converges to this element u B as /i — 0 and, in addition, the order of accuracy of scheme (21) coincides with the order... 

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