Model truncation error

Statistical and algebraic methods, too, can be classed as either rugged or not they are rugged when algorithms are chosen that on repetition of the experiment do not get derailed by the random analytical error inherent in every measurement,i° 433 is, when similar coefficients are found for the mathematical model, and equivalent conclusions are drawn. Obviously, the choice of the fitted model plays a pivotal role. If a model is to be fitted by means of an iterative algorithm, the initial guess for the coefficients should not be too critical. In a simple calculation a combination of numbers and truncation errors might lead to a division by zero and crash the computer. If the data evaluation scheme is such that errors of this type could occur, the validation plan must make provisions to test this aspect. 

In dealing with the SGS terms, Revstedt et al. (1998, 2000) and Revstedt and Fuchs (2002) did not use any model rather, they assumed these terms were just as small as the truncation errors in the numerical computations. This heuristic approach lacks physics and does not deserve copying. A most welcome aspect of LES is that the SGS stresses may be conceived as being isotropic, i.e., insensitive to effects of the larger scales, to the way the turbulence is induced and to the complex and varying boundary conditions of the flow domain. Exactly this... 

In section 4, we established that the orbital truncation error represents a serious obstacle to the accurate calculation of AEs. Next, in section 5, we found that this problem may be solved in two different ways we may either employ wave functions that contain the interelec-tronic distance explicitly (in particular the R12 model), or we may try to extrapolate to the basis-set limit using energies obtained with finite basis sets. In the present section, we shall apply both methods to a set of small molecules, to establish whether or not these techniques are useful also for systems of chemical interest. 

A. Lagrangian Framework. An ideal subgrid model should be constructed on a Lagrangian hydrodynamics framework moving with the macroscopic flow. This requirement reduces purely numerical diffusion to zero so that realistic turbulence and molecular mixing phenomena will not be masked by non-physical numerical smoothing. This requirement also removes the possibility of masking purely local fluctuations by truncation errors from the numerical representation of macroscopic convective derivatives. 

The accuracy of the averaged model truncated at order p9(q 0) thus depends on the truncation of the Taylor series as well as on the truncation of the perturbation expansion used in the local equation. The first error may be determined from the order pq 1 term in Eq. (23) and may be zero in many practical cases [e.g. linear or second-order kinetics, wall reaction case, or thermal and solutal dispersion problems in which / and rw(c) are linear in c] and the averaged equation may be closed exactly, i.e. higher order Frechet derivatives are zero and the Taylor expansion given by Eq. (23) terminates at some finite order (usually after the linear and quadratic terms in most applications). In such cases, the only error is the second error due to the perturbation expansion of the local equation. This error e for the local Eq. (20) truncated at 0(pq) may be expressed as... 

In this second method, we look for the numerical solution of the ideal model by ignoring the RHS of Eq. 10.61. In the process we introduce a truncation error which we adjust in such a way that this numerical error accoxmts as closely as possible for the RHS of Eq. 10.61 which is initially ignored. If we ignore the RHS term of Eq. 10.61, it becomes... 

This equation is equivalent to Eq. 10.71 with Da = 0, which is expected because we are now writing the Lax-Wendroff equation for solving Eq. 10.72, which is equivalent to Eq. 10.61 with Da = 0. However, in this chapter we are interested in solving the equilibrium-dispersive model (Eq. 10.61), not the ideal model (Eq. 10.72). So, neither the Lax-Wendroff nor any similar scheme which gives a high-order truncation error is suitable for our purpose. The error made is too small to account for the dispersive effects in an actual column. 

Chapter 10, which provides satisfactory accuracy and is the simplest and fastest calculation procedure. This method consists of neglecting the second-order term (RHS of Eq. 11.7) and calculating numerical solutions of the ideal model, using the numerical dispersion (which is equivalent to the introduction in Eq. 11.7 of a first-order error term) to replace the neglected axial dispersion term. Since we know that any finite difference method will result in truncation errors, the most effective procedure is to control them and to use them to simplify the calculation. The results obtained are excellent, as demonstrated by the agreement between experimental band profiles recorded with single-component samples and profiles calculated [2-7]. Thus, it appears reasonable to use the same method in the calculation of solutions of multicomponent problems. However, in the multicomponent case a new source of errors appears, besides the errors discussed in detail in Chapter 10 (Section 10.3.5). 

Several schemes and algorithms for solving the fluid dynamic part of the model have been published. This work has been concentrated on several items. Most important, one avoids using the very diffusive first order upwind schemes discretizing the convective terms in the multi-fluid transport equations. Instead higher order schemes that are more accurate have been implemented into the codes [62, 139, 140, 65, 105, 66[. The numerical truncation errors induced by the discretization scheme employed for the convective terms may severely alter the numerical solution and this can destroy the physics reflected... 

A proper model validation procedure consists of a model verification part and a part where the model predictions are compared to experimental data [61]. The model verification may be performed by the method of manufactured solutions[14 7, 163]. The method of manufactured solutions consists in proposing an analytical solution, preferably one that is infinitely differentiable and not trivially reproduced by the numerical approximation, and the produced residuals are simply treated as source terms that produce the desired or prescribed solution. These source terms or residuals are referred to as the consistent forcing functions. This method can be used to confirm that there are no programming errors in the code and to monitor the truncation error behavior during the iteration process. 

A consistent numerical scheme produces a system of algebraic equations which can be shown to be equivalent to the original model equations as the grid spacing tends to zero. The truncation error represents the difference between the discretized equation and the exact one. For low order finite difference methods the error is usually estimated by replacing all the nodal values in the discrete approximation by a Taylor series expansion about a single point. As a result one recovers the original differential equation plus a remainder, which represents the truncation error. 

An important aspect of Eulerian reactor models is the truncation errors caused by the numerical approximation of the convection/advection terms [82], Very different numerical properties are built into the various schemes proposed for solving these operators. The numerical schemes chosen for a particular problem must be consistent with and reflect the actual physics represented by the model equations. 

The mole fractions at the end of the time increment (equivalent to the distillate increment) are calculated by numerical integration. The magnitude of the time increment is determined by stability and truncation considerations. The batch distillation model contains tray holdups with time constants much smaller than the reboiler time constant. These conditions ( stiff systems ) can cause computational instability unless very small time increments are used. The penalty is excessive computing time and the likelihood of incurring truncation errors. Distefano (1968) provides values for the maximum time increment size consistent with stability for a number of integration schemes. The same time increment is used to determine the incremental distillate rate for the flrst step. 

The majority of quantum-chemistry calculations have been carried out by employing the independent particle model in the framework of the HF method. In the most widely used approach molecular orbitals are expanded in predefined one-particle basis functions which results in recasting the integro-differential HF equations into their algebraic equivalents. In practice, however, the basis set used is never complete and very often far too limited to describe essential features of HF orbitals, for example, their behaviour in the vicinity of nuclei. That is why such calculations always suffer from the so called basis set truncation error . This error is difficult to estimate and often leads to low credibility of the results. 

Notice that the approximation is exact, but that The reason is that the residual in Eq. (7.93) includes the truncation error for the approximation plus the residual error term, s. Equation (7.93) can be thought of as a linear model... 

The convergence behaviour observed in the present study for continuum models of solvated species using systematic sequences of even-tempered basis sets of Gaussian primitive functions of s- and p- symmetry mirrors that observed for the gas phase molecular species. Gas phase matrix Hartree-Fock calculations for the isoelectronic molecules studied in the present work which approach a suh-pHartreelevel of accuracy for the total energy have been reported elsewhere[18]-[25]. The present work demonstrates that this approach can be applied to molecules in solution by making use of continuum models. The basis set truncation error would then be 3 orders of magnitude smaller than the AE values. 

This error emerges at the replacement of all derivatives by finite differences both over the coordinates of space and time. However, its effect is especially noticeable in accounting for the influence of mass-transport. The truncation error as a result of computation associated with mass-transport is analogous to hydrodynamic dispersion and for this reason is often called numerical dispersion. It exists only in computation and is caused by the fact that the distance actually run over by water during the accepted time step At is not exactly equal to the accepted for computation step in distance of Ax. That is why at computation in cells (blocks), beside natural mixing associated with hydrodynamic dispersion D, appears also erroneous dispersion caused by the mismatch between the accepted value of the time step and the distance step. A result is that the final modeling results reflect not only hydrodynamic but also numerical dispersion, i.e., not D. but D + where the second addend is numerical dispersion coefficient. 

The Leonard stress term can be computed directly and does not need to be modeled. However, as shown by Shaanan et al. [50], the Leonard stresses are of the same order as the truncation errors of a second-order discretization scheme, and therefore, L can be implicitly accounted for. Consequently, a computed solution may depend on the numerical scheme that is used. 

The effort to develop a neutron transport code based on an improved nodal method has been continued in order to treat the Hex-Z geometry of FBR cores more accurately. In order to reduce truncation error of the code, a new method to treat the radial leakage has been developed, in which the distribution of node boundary fluxes is obtained from local two-dimensional flux distribution. The local flux distribution is evaluated from average fluxes at surrounding nodes and node boundaries by second-order polynomials. An FBR core model with extremely large-sized assemblies was calculated by the new method and the results were compared with those of a reference Monte Carlo calculation. While the previous method overestimated the criticality by 0.3% dk for a control rod-insertion case, the new method agreed well with the Monte Carlo result. 

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