Order accuracy

In other chapters, the order of accuracy of various methods is referred to. Here this concept is defined and two methods of calculating the order are 

A simulation results in a number (or a vector of numbers) at some time. Depending on the dimensionality of the problem, the simulation uses intervals in time ST and one or more space intervals. Often there is only one space interval, here given the symbol H. A result - a current, or a concentration, for example - will, due to truncation errors, have an error associated with it, that can be expressed in the following way. The discussion is, for the moment, restricted to an ode with interval size h. Then the simulated result at time t can be written as a polynomial 

The number p is of great interest, more so than the constants, which are generally unknown and are usually unimportant (except in rare cases) when deciding on a given method. This is because a high order accuracy means that if we decrease h, we dramatically improve the accuracy. Conversely, this is not the case for a small p. So, first-order methods such as EX or BI mean that we must decrease the intervals greatly in order to achieve some 

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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 — solves problem (38) with the right-hand side V ft (and... 

Exact and truncated schemes are quite applicable in the estimation of the accuracy of schemes (16)-(17) in Section 2. This approach allows to weaken or get rid of the smoothness of the functions k, q, f involved in the estimation of the accuracy order of schemes (16)-(17) in Section 2. 

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). 

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. 

The two methods that stand out in terms of efficiency and convenience are BDF and extrapolation. Both require minimal programming effort, and can be extended to higher-order spatial derivatives. However, in the case of BDF, a limit is encountered. For the most convenient start-up methods such as the simple or the rational start, the accuracy from BDF is limited to 0(8T2). This means for one thing that one need not go beyond 3-point BDF (which is 0 8T2) in itself), but that no marked improvement can be gained from higher-order spatial derivative approximations, because there will then be a mismatch between the accuracy orders with respect to the time and spatial intervals. 

Note that (Fig. 14.11) both curves show similar accuracy at small Nt- If one measures the gradient there, it verifies the expectation of an accuracy order of 0(6T2). However, extrapolation then shows a sharp dip, followed by an approach to a constant error. The dip is due to the actual error crossing the zero line. BDF simply flattens out and approaches the same constant error. 

Mass spectrometry can determine the molecular weights of peptides and proteins with mass accuracies orders of magnitude better than the molecular weights determined by gel electrophoresis. It is important to note that in determining molecular... 

Recently, we have discovered a general method for solving the SE [41-47]. The free-complement (FC) method [45] is the most popular method. A number of highly accurate results have been obtained for various atoms and molecules [41-58]. This was possible because the FC wave functions converge to the exact wave functions as the accuracy (order) is increased. An important feature is that the complement functions, which are the elements of the FC wave function, are automatically produced by applying the Hamiltonian to a simple initial wave function [45]. Therefore, this method can be applied to any system where the analytical form of the Hamiltonian is known. 

Cash and receivables Order cycle time Order completion rate Invoice accuracy Order shipment performance Receivable/credits caused by system malfunction... 

Order entry There are two types of orders at Valvex inbound orders and outbound orders. The system generates inbound orders based on purchase orders sent by the ERP system. It also generates outbound orders based on production orders or customer orders transmitted by the ERP system. In order to ensure order data accuracy, orders are entered electronically into the system. 

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