Stability errors

The heightened appreciation of resonance problems, in particular, has been quite recent [63, 62], and contrasts the more systematic error associated with numerical stability that grows systematically with the discretization size. Ironically, resonance artifacts are worse in the modern impulse multiple-timestep methods, formulated to be symplectic and reversible the earlier extrapolative variants were abandoned due to energy drifts. 

To obtain the unconditional stability of the midpoint method in local coordinates, one would have to consider the decoupling transformation from cartesian to local coordinates for the discrete variables etc. But this transformation, which for the continuous variables is not constant, necessarily is in error which depends on k, not e. The stability properties of the discrete dynamical systems obtained by the midpoint discretization in the different sets of coordinatc.s may therefore be significantly different when it 3> e [3]. 

In order to compare the efficiency of the SISM with the standard LFV method, we compared computational performance for the same level of accuracy. To study the error accumulation and numerical stability we monitored the error in total energy, AE, defined as... 

These various techniques were recently applied to molecular simulations [11, 20]. Both of these articles used the rotation matrix formulation, together with either the explicit reduction-based integrator or the SHAKE method to preserve orthogonality directly. In numerical experiments with realistic model problems, both of these symplectic schemes were shown to exhibit vastly superior long term stability and accuracy (measured in terms of energy error) compared to quaternionic schemes. 

In experiments, the two symplectic methods ROT and SPL performed very similarly in terms of error propagation and long term stability. The ex-... 

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]). 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

Spike recoveries for samples are used to detect systematic errors due to the sample matrix or the stability of the sample after its collection. Ideally, samples should be spiked in the field at a concentration between 1 and 10 times the expected concentration of the analyte or 5 to 50 times the method s detection limit, whichever is larger. If the recovery for a field spike is unacceptable, then a sample is spiked in the laboratory and analyzed immediately. If the recovery for the laboratory spike is acceptable, then the poor recovery for the field spike may be due to the sample s deterioration during storage. When the recovery for the laboratory spike also is unacceptable, the most probable cause is a matrix-dependent relationship between the analytical signal and the concentration of the analyte. In this case the samples should be analyzed by the method of standard additions. Typical limits for acceptable spike recoveries for the analysis of waters and wastewaters are shown in Table 15.1. ... 

Internal standards at a known concentration are added to the sample after its preparation but prior to analysis to check for GC retention-time accuracy and response stability. If the internal standard responses are in error by more than a factor of two, the analysis must be stopped and the initial calibration repeated. Only if all the criteria have been met can sample analysis begin. 

Until now we have been discussing the kinetics of catalyzed reactions. Losses due to volatility and side reactions also raise questions as to the validity of assuming a constant concentration of catalyst. Of course, one way of avoiding this issue is to omit an outside catalyst reactions involving carboxylic acids can be catalyzed by these compounds themselves. Experiments conducted under these conditions are informative in their own right and not merely as means of eliminating errors in the catalyzed case. As noted in connection with the discussion of reaction (5.G), the intermediate is stabilized by coordination with a proton from the catalyst. In the case of autoprotolysis by the carboxylic acid reactant, the rate-determining step is probably the slow reaction of intermediate [1] ... 

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. 

Without stabilization, the step response of the roll dynamies produees a 45% overshoot and a settling time of 10 seeonds. The stabilization eontrol system is required to provide a step response with an overshoot of less than 25%, a settling time of less than 2 seeonds, and zero steady-state error. 

Robust stability provides a minimum requirement in an environment where there is plant model uneertainty. For a eontrol system to have robust performanee it should be eapable of minimizing the error for the worst plant (i.e. the one giving the largest error) in the family G(jtu) [Pg.308]

Thus, the user can input the minimum site boundary distance as the minimum distance for calculation and obtain a concentration estimate at the site boundary and beyond, while ignoring distances less than the site boundary. If the automated distance array is used, then the SCREEN model will use an iteration routine to determine the maximum value and associated distance to the nearest meter. If the minimum and maximum distances entered do not encompass the true maximum concentration, then the maximum value calculated by SCREEN may not be the true maximum. Therefore, it is recommended that the maximum distance be set sufficiently large initially to ensure that the maximum concentration is found. This distance will depend on the source, and some trial and error may be necessary however, the user can input a distance of 50,000 m to examine the entire array. The iteration routine stops after 50 iterations and prints out a message if the maximum is not found. Also, since there may be several local maxima in the concentration distribution associated with different wind speeds, it is possible that SCREEN will not identify the overall maximum in its iteration. This is not likely to be a frequent occurrence, but will be more likely for stability classes C and D due to the larger number of wind speeds examined. 

The determination of the critical GLC is a trial and error computation of GLC s due to various wind speeds, atmospheric stabilities and downwind distances. The maximum value obtained from these procedures is the critical GLC. Because of the number of computations involved, calculations should be performed on the computer. Software simulation is also necessary to calculate GLC s due to multiple stack cases. Wind direction is an additional variable that must be taken into account with multiple stact cases. 

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. 

From the calibration point of view, manometers can be divided into two groups. The first, fluid manometers, are fundamental instruments, where the indication of the measured quantity is based on a simple physical factor the hydrostatic pressure of a fluid column. In principle, such instruments do not require calibration. In practice they do, due to contamination of the manometer itself or the manometer fluid and different modifications from the basic principle, like the tilting of the manometer tube, which cause errors in the measurement result. The stability of high-quality fluid manometers is very good, and they tend to maintain their metrological properties for a long period. 

Rawlings etal. (1992) analysed the stability of a eontinuous erystallizer based on the linearization of population and solute balanee. Their model did not depend on a lumped approximation of partial differenee equations and sueeess-fully predieted the oeeurrenee of sustained oseillations. They demonstrated that simple proportional feedbaek eontrol using moments of CSD as measurements ean stabilize the proeess. It was eoneluded that the relatively high levels of error in these measurements require robust design for effeetive eontrol. 

The inherent error between true vapor pressure and RVP means that a stabilizer designed to produce a bottoms liquid with a true vapor pressure equal to the specified RVP will be conservatively designed. The a[>or pressures of various hydrocarbon components at 100°F are given in Table 6-1. 

The six-membered rings 8.12a and 8.12b adopt chair conformations with all three halogen atoms in axial positions. This arrangement is stabilized by the delocalization of the nitrogen lone pair into an S-X a bond (the anomeric effect) All the S-N distances are equal within experimental error [ld(S-N)l = 1.60 (8.12a)/ 1.59 A (8.12b) ]. 

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