Error and Stability

Normally, the truncation error goes to zero as h approaches zero. Hence, we can make the error as small as we want by choosing an appropriate h. However, the smaller the h, the larger the rounding error on the computed solution. In practice for a fixed word length in the computer arithmetic, there will be a size h, below which the rounding error will become the dominant contribution to the overall error. Because a faster solution is always preferred, it is desirable to choose as big a step size as possible where the truncation error is predominant. 

For a method to be convergent, the finite difference solution must approach the true solution as the interval or the step size approaches zero. 

The concept of stability is associated with the propagation of errors of the numerical integration technique as the calculation progresses with a finite interval size, and is related to the effects of errors made in a single step on succeeding steps. The problem of stability arises because in most instances the order m of the approximate difference equation (Equation 2.4) is higher than the original differential equation q (Equation 2.1), whose actual solution can be written as m q)  

the difference equation may contain extraneous terms that may dominate the equation and bear little or no resemblance to the true solution. It happens frequently that the spurious solutions do not vanish even in the limits as the increment size approaches zero. This phenomenon is called strong 

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Computational issues that are pertinent in MD simulations are time complexity of the force calculations and the accuracy of the particle trajectories including other necessary quantitative measures. These two issues overwhelm computational scientists in several ways. MD simulations are done for long time periods and since numerical integration techniques involve discretization errors and stability restrictions which when not put in check, may corrupt the numerical solutions in such a way that they do not have any meaning and therefore, no useful inferences can be drawn from them. Different strategies such as globally stable numerical integrators and multiple time steps implementations have been used in this respect (see [27, 31]). 

In addition to stability considerations, the order of the approximation error is also a function of the system index. From the results of Brenan and Petzold (1987), systems of equations of higher index can be considered simply by choosing the appropriate IRK method with the appropriate integration error constraints. Based on error and stability considerations, Logsdon and Biegler (1989) concluded that minimum order requirements for collocation methods are the following ... 

Dassl, solves stiff systems of differential-algebraic equations (DAE) using backward differentiation techniques [13,46]. The solution of coupled parabolic partial differential equations (PDE) by techniques like the method of lines is often formulated as a system of DAEs. It automatically controls integration errors and stability by varying time steps and method order. 

Determining model robustness through sensitivity analysis, examination of parametric/nonparametric standard errors, and stability testing with or without predictive performance depending on the objective of the PMKD. 

DISPERSION ERROR AND STABILITY ANALYSIS 3.5.1 The Enhanced Dispersion Relation... 

T. Kashiwa, H. Kudo, Y. Sendo, T. Ohtani, and Y. Kanai, The phase velocity error and stability condition of the three-dimensional nonstandard FDTD method, IEEE Trans. Magn., vol. 38, no. 2, pp. 661-664, Mar. 2002.doi 10.1109/20.996172... 

The replicates at the center point have two main objectives providing a measure of pure error and stabilizing the variance of the predicted response. To stabilize the variance, a rule of thumb is to make 3-5 repeated runs if a is close to Vfe, and only one or two if it is close to 1. Between these extremes, 2-4 replicates are indicated. To obtain a more... 

Numerical integration techniques are necessary in modeling and simulation of batch and bio processing. In this chapter we described error and stability criteria for numerical techniques. Various numerical techniques for solution of stiff and non-stiff problems are discussed. These methods include one-step and multi-step explicit methods for non-stiff and implicit methods for stiff systems, and orthogonal collocation method for ordinary as well as partial differential equations. These methods are an integral part of some of the packages like MATLAB. However, it is important to know the theory so that appropriate method for simulation can be chosen. 

As can be seen in the table above, the upper two results for heat transfer coefficients hp between particle and gas are about 10% apart. The lower three results for wall heat transfer coefficients, h in packed beds have a somewhat wider range among themselves. The two groups are not very different if errors internal to the groups are considered. Since the heat transfer area of the particles is many times larger than that at the wall, the critical temperature difference will be at the wall. The significance of this will be shown later in the discussion of thermal sensitivity and stability. 

Contact temperature measurement is based on a sensor or a probe, which is in direct contact with the fluid or material. A basic factor to understand is that in using the contact measurement principle, the result of measurement is the temperature of the measurement sensor itself. In unfavorable situations, the sensor temperature is not necessarily close to the fluid or material temperature, which is the point of interest. The reason for this is that the sensor usually has a heat transfer connection with other surrounding temperatures by radiation, conduction, or convection, or a combination of these. As a consequence, heat flow to or from the sensor will influence the sensor temperature. The sensor temperature will stabilize to a level different from the measured medium temperature. The expressions radiation error and conduction error relate to the mode of heat transfer involved. Careful planning of the measurements will assist in avoiding these errors. 

The question of the accuracy of the scheme, being of principal importance in the theory, amounts to studying the error of approximation and stability of the scheme. Stability analysis neces.sitates imposing a priori estimates for the difference problem solution in light of available input data. This is a problem in itself and needs investigation. 

It can easily be shown that for the upwind scheme all coefficients a appearing in Eq. (37) are positive [81]. Thus, no unphysical oscillatory solutions are foimd and stability problems with iterative equation solvers are usually avoided. The disadvantage of the upwind scheme is its low approximation order. The convective fluxes at the cell faces are only approximated up to corrections of order h, which leaves room for large errors on course grids. 

Important thermodynamic properties that relate to the structure and stability of the chalcogen ailotropes and their polyatomic cations are the formation enthalpies listed in Table 2. Only reliable experimentally or quantum chemically established numbers have been included. From Table 2 it is evident that tellurium is the least investigated with respect to the entries thus, there is clearly space for more thorough experimental or quantum chemical work in this direction. Therefore, we have assessed the missing Te data from the IP determination in ref. 12 (PE spectroscopy) and ref. 13 (quantum chemical calculations) and have put them in the table in parentheses, although it is clear that the associated error bars are relatively high. The data in ref. 14 were not considered. 

For those scientists who had to perform quantitation, the linearity of the A/D was also critical. Linearity is the condition in which the detector s response is directly proportional to the concentration or amount of a component over a specified range of component concentrations or amounts. It is imperative that the A/D not add any additional error or variability to the performance of the detector. The resulting calibration curve now becomes dependent on the combined linearity of the detector and the /VD. Accurate quantitation requires that the system is linear over the range of actual sample concentrations or amounts. Many pharmaceutical assays, like degradation and stability studies, require that the system be able to identify and quantitate very disparate levels of peaks. In many cases, this translates into a 3 to 4 order of magnitude difference between the main active component and the impurities that need to be quantitated. 

Figures 57 and 58 shows the estimation results for the intervals of the unmeasured states Cti and Z. Notice how the interval bounds estimated by the interval observer envelop correctly these unmeasured states. For all the other unmeasured states, notice that although the interval observer design did not allow us to tune the convergence rate, the interval observer showed excellent robustness and stability properties and provided satisfactory estimation results in the event of highly corrupted measurements and operational failures. Notice in particular, the robustness of the interval observer around day 25 when the inlet concentrations drastically increased and when a major disturbance occurred at day 31, due to an operational failure, resulting in a rapid fall of both, the dilution rate (which actually fell to zero) and the substrate concentration readings. Off-line readings of Cti and Z (not used in the state estimation calculations) were also added to validate the proposed interval observer design (see Figures 57 and 58). It should be noticed that the compromise between the convergence rate and robustness was not fully achieved until the estimation error dynamics reached the steady state.

Accurate measurements of the frequency-resolved transverse spin relaxation T2) of Rb NMR on single crystals of D-RADP-x (x = 0.20, 0.25, 0.30, 0.35) have been performed in a Bq field of 7 Tesla as a function of temperature. The probe head was placed in a He gas-flow cryostat with a temperature stability of 0.1 K. To obtain the spin echo of the Rb - 1/2 -o-+ 1/2 central transition we have used the standard (90 - fi - 180y -ti echo - (2) pulse sequence with an appropriate phase-cycling scheme to ehminate quadrature detection errors and unwanted coherences due to pulse imperfections. To avoid sparking in the He gas, the RF-field Bi had to be reduced to a level where the 7T/2-pulse length T90 equalled 3.5 ps at room temperature. 

The onset of the HOMO, homo (leading edge of the peak), is found at 1.8 eV below E. This value is very close to opt — 2.2 eV (see Table 1.6). The change in the sample (/>m, monitored by the shift of the secondary electron cutoff, is shown as inset in Fig. 4.24(a). For increasing pentacene coverage, the onset moves abruptly toward higher kinetic energy and stabilizes (within experimental error) c. 0.3 eV above the iiutial value for a pentacene thickness of 0.5 nm. For pentacene the experimentally obtained value of /e is 4.9 eV. 

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