Local time error

The algorithms discussed earlier for time averaging and local time stepping apply also to velocity, composition PDF codes. A detailed discussion on the effect of simulation parameters on spatial discretization and bias error can be found in Muradoglu et al. (2001). These authors apply a hybrid FV-PDF code for the joint PDF of velocity fluctuations, turbulence frequency, and composition to a piloted-jet flame, and show that the proposed correction algorithms virtually eliminate the bias error in mean quantities. The same code... 

The data can then be downloaded directly into the clinical trial database. The main drawbacks include the need for training the subject in the diary s use, possible errors in local time settings and the logistics of distribution, maintenance and recovery of the diaries. 

High-level DAE software (e.g., Dassl) makes a time-step selection based on an estimate of the local truncation error, which depends on the difference between a predictor and a corrector step [13,46]. If the difference is too great, the time step is reduced. In the limit of At 0, the predictor is just the initial condition. For the simple linear problem illustrated here, the corrector will always converge to the correct solution y2 = 1, independent of the time step. However, if the initial condition is y2 1, then there is simply no time step for which the predictor and corrector values will be sufficiently close, and the error estimate will always fail. Based on this simple problem, it may seem like a straightforward task to build software that identifies and avoids the problem, and there is current research on the subject [13], The problem is that in highly nonlinear, coupled, problems the inconsistent initial conditions can be extremely difficult to identify and fix in a general way. 

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. 

Analysis of the algorithm requires understanding the behavior of local truncation error. We use the nomenclature y(tn) to mean the exact analytic solution evaluated at some time tn and yn to represent the numerical solution at tn. It is clear that the true analytic solution must satisfy the differential equation everywhere,... 

The first three terms represent the forward Euler algorithm operating on the exact solution, with the last term [in square brackets] providing a measure of the local truncation error. The local truncation error can be identified through a Taylor series expansion of the solution about the time tn ... 

In addition to the restrictions that stability places on the step size hn, we also need to be concerned with how accuracy affects the choice of step size. Assume that the local accuracy is to be controlled to within a certain tolerance e and that accuracy can be estimated by the local truncation error. The time step must be chosen to keep a norm of the local truncation error below the tolerance, that is ... 

In regions where the solution varies slowly, accuracy considerations alone would permit a large time step. However, for a stiff problem where nearby solutions vary rapidly, stability demands a very small time step, even in regions where the solution is changing slowly and the local truncation error (accuracy) can be controlled easily with a large time step. 

Vode, solves stiff systems of ordinary differential equations (ODE) using backward differentiation techniques [49]. It implements rigorous control of local truncation errors by automatic time-step selection. It delivers computational efficiency by automatically varying the integration order. 

Fig. 19 Results of a diet study of surface-water ferrozine-reactive iron concentrations while following a body of water in the Gulf of Carpentaria, north of Australia. Plot A shows the light intensity measured over the day with the first point being at sunrise and the last at sunset. Plot B depicts filterable (< 0.4 pm) ferrozine-reactive iron concentrations for the same local time period. Plot C shows particulate ( ) and total ( ) ferrozine-reactive iron concentrations. Error bars indicate 98% confidence limits (from [153])...

Consider the variation of the solution of a transient heat transfer problem with time at a specified nodal point. Both the numerical and actual (exact) solutions coincide at the beginning of the first time step, as expected, but the numerical. solution deviates from the exact. solution as the time t increases. The difference between the two solutions at t Ar is due to the approximation at the first time step only and is called the local discretization error. One would expect the situation to get worse with each step since the second step uses the erroneous result of the first step a,s its starting point and adds a second local discretization error on top of it, as shown... 

To have an idea about the magnitude of the local discretization error, consider the Taylor serie.s expansion of the temperature at a specified nodal point m about time... 

Paper subject diaries are notoriously poor in their legibility, completeness and accuracy. Electronic diaries are small, portable de vices that can present text and graphics to the subject. They allow the subject to record and store responses, which can be time-stamped for each entry made. The data can then be downloaded directly into the clinical trial database. The main drawbacks include the need for training of the subject in the diary s use, possible errors in local time settings and the logistics of distribution, maintenance and recovery of the diaries. 

This formula is simply the explicit Euler method. The local truncation error is of order of Oih ), that is, the error is proportional to if all the previous values, y , y i,... are exact. However, in the integration from time t = 0, the error of the method at time is accurate up to Oih) because the number of integration step is inversely proportional to h. 

Equation (3.1.5) can therefore be used to compute the value a at a new time, t + At, based upon a given value of the function f x, t) which is evaluated at the old time t. Since we know the initial condition at t = 0, we can start this forward -marching scheme and proceed out to any value of t in steps of At. This numerical algorithm of equation (3.1.5) is known as Euler s method. Unfortunately, the local truncation error for this algorithm is large and is of the order of At. To maintain accuracy with a truncation error of this size, the step size needs to be very small. In practice other methods of greater accuracy have been devised in order to allow for increased step sizes. 

For this new produced method the authors has given a complete local truncation error analysis and stability analysis. From the error analysis the authors have proved that for the Classical Six-Step Method the error increases as the fifth power of G. For the Trigonometrically-Fitted Method produced in this paper the error increases as the fourth power of G. So, for the numerical solution of the time independent radial Schrodinger equation the Trigonometrically-fitted Six-Step Method produced in this paper is the most accurate one, especially for large values of G = Vc E. From the stability analysis the authors have proved that the new proposed method is P-stable. The numerical results verify the theoretical results presented in the paper. 

In the representation of the local squeeze velocity, an embedded Tr.ipezoidal rule was employed for the pressure values from the current time step. The accuracy of this representation was influenced by the spacing of the discrete pressure distribution which was not controlled within the DGEAR library routine. Thus, reliability checks had to be performed by varying the level of discretization. The level of discretization also influenced the accuracy of the calculated load. The reliability of the part of the solution which was controlled by DGEAR had checks made by varying the allowable local truncation error within DGEAR. 

As the local error is the same as in the forward difference method, i.e. 0 k) + OQp-), the error from the time discretization dominates, assuming step sizes and h k. Although the method is unconditionally stable, its accuracy is low due to the large local truncation error. Overall, the backward difference method is 0 k), which means that the error at a certain point decreases linearly with k. In other words, if k is cut in half, so is the error. In contrast, if h is cut in half (for a given K), the error would be about the same, and the amount of computation would increase. 

Cases 9 and 10 illustrate the type of convergence study which is possible with methods in which the local approximation error has a known parameter dependence. In case 9 the error should be at least a factor of (1.5) = 5 times larger than case 10. Thus one can confidently and conservatively ascribe RHF-limit accuracy to the digits that are stable in the two calculations. That is, in case 10 the uncertainty should be no greater than the fifth decimal place for the total and orbital energies and the third-fourth for the moments. 

The only other simple local error criterion that can be applied is to require that the local relative error remain within some bound. The solution variables will always have some local values and the important local criterion with respect to the accuracy of a solution is how the error in the solution compares with the local values of the solution variables. But what about a problem such as that mentioned above where the initial boundary condition on all variables is zero It appears that there are no finite solution values with which to compare the error. However, after having taken the first step in a solution the variables will have some finite values even if the values may be very small. Thus a comparison can be made of an estimated error with the average of the solution variables over a possible time step. 

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