Error numerical methods, controlling

Overview Reconciliation adjusts the measurements to close constraints subject to their uncertainty. The numerical methods for reconciliation are based on the restriction that the measurements are only subject to random errors. Since all measurements have some unknown bias, this restriction is violated. The resultant adjusted measurements propagate these biases. Since troubleshooting, model development, ana parameter estimation will ultimately be based on these adjusted measurements, the biases will be incorporated into the conclusions, models, and parameter estimates. This potentially leads to errors in operation, control, and design. 

The successful numerical solution of differential equations requires attention to two issues associated with error control through time step selection. One is accuracy and the other is stability. Accuracy requires a time step that is sufficiently small so that the numerical solution is close to the true solution. Numerical methods usually measure the accuracy in terms of the local truncation error, which depends on the details of the method and the time step. Stability requires a time step that is sufficiently small that numerical errors are damped, and not amplified. A problem is called stiff when the time step required to maintain stability is much smaller than that that would be required to deliver accuracy, if stability were not an issue. Generally speaking, implicit methods have very much better stability properties than explicit methods, and thus are much better suited to solving stiff problems. Since most chemical kinetic problems are stiff, implicit methods are usually the method of choice. 

Besides examining these properties of numerical methods, specific efforts need to be made to assess the accuracy of numerical solutions of flow processes. Various types of errors and possible ways of estimating and controlling these errors are discussed... 

Often, the step size is chosen to increase the accuracy of the numerical solution. More importantly, the step size controls not only the error, but also the stability of the numerical scheme. The following problem is selected to illustrate the stability characteristics of the numerical method ... 

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. 

The thick sohd Hne is what would occur if 6 perfectly controlled the error in the reaction probabilities. The solid line with squares is the average of the error, and the dashed line with circles is the maximum error from the systems studied all as a function of S. We see the remarkable result that, even in the worst case of maximum error, 6 reliably controls the error in the reaction probability. Thus, even if we were studying a system in resonance (i.e, a small effective absorption rate F) requiring a larger TabCi would not change. This kind of control is an important aspect of any numerical method, i.e. that one be able to determine a priori how accurate the calculation is and consequently how much computational effort is required. 

Numerical methods are incapable of solving nonlinear equations explicitly and the actual behavior must be approximated by a sequence of linear steps. Incremental and iterative methods are available to solve a system of nonlinear equations. In the incremental method, the response is approximated by dividing the solution into a number of linear increments and updating the stiffness at each increment. The incremental method may underestimate the nonlinear behavior and a progressive divergence from the actual response may be observed. A better approximation would be obtained by decreasing the size of the increments but for a controlled reduction in error an iterative method is required. 

The reactivity computer measures real time variation of reactivity. It can be used virtually with all neutron detectors for the reactor power monitoring simulataneously. In order to cover the startup channels, numerical methods for inverse point kinetics are accutely reviewed and tested in detail to verify error trend and to search the best algorithm. The program made for the test itself can be utilized for the simulation of other reactivity measurements to verify source of error and its trend - such as the gamna background effect when an uncompensated ionization chamber is used, the source effect, conventional rod drop method, etc. The reactivity computer can be used at any reactor power level from startup to power range. So far, it has been utilized for the control rod calibration, reactivity coefficient measurement, etc. 

In the previous section we described several internal methods of quality assessment that provide quantitative estimates of the systematic and random errors present in an analytical system. Now we turn our attention to how this numerical information is incorporated into the written directives of a complete quality assurance program. Two approaches to developing quality assurance programs have been described a prescriptive approach, in which an exact method of quality assessment is prescribed and a performance-based approach, in which any form of quality assessment is acceptable, provided that an acceptable level of statistical control can be demonstrated. 

Becker, R., Rannachee, R., A feedback approach to error control infinite element methods basic analysis and examples, East-West J. Numer. Math. 4 (1996) 237-264. 

CFD may be loosely thought of as computational methods applied to the study of quantities that flow. This would include both methods that solve differential equations and finite automata methods that simulate the motion of fluid particles. We shall include both of these in our discussions of the applications of CFD to packed-tube simulation in Sections III and IV. For our purposes in the present section, we consider CFD to imply the numerical solution of the Navier-Stokes momentum equations and the energy and species balances. The differential forms of these balances are solved over a large number of control volumes. These small control volumes when properly combined form the entire flow geometry. The size and number of control volumes (mesh density) are user determined and together with the chosen discretization will influence the accuracy of the solutions. After boundary conditions have been implemented, the flow and energy balances are solved numerically an iteration process decreases the error in the solution until a satisfactory result has been reached. 

Reliable process data are the key to the efficient operation of chemical plants. With the increasing use of on-line digital computers, numerous data are acquired and used for on-line optimization and control. Frequently these activities are based on small improvements in process performance, but it must be noted that errors in process data, or inaccurate and unreliable methods of resolving these errors, can easily exceed or mask actual changes in process performance. 

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