QUICK scheme

Hunt and Kulmala have solved the full turbulent fluid flow for the Aaberg system using the k-e turbulent model or a variation of it as described in Chapter 13— the solution algorithm SIMPLE, the QUICK scheme, etc. Both commercial software and in-house-developed codes have been employed, and all the investigators have produced very similar findings. 

It should also be remembered that the discretization scheme influences the accuracy of the results. In most CFD codes, different discretization schemes can be chosen for the convective terms. Usually, one can choose between first-order schemes (e.g., the first-order upwind scheme or the hybrid scheme) or second-order schemes (e.g., a second-order upwind scheme or some modified QUICK scheme). Second-order schemes are, as the name implies, more accurate than first-order schemes. However, it should also be remembered that the second-order schemes are numerically more unstable than the first-order schemes. Usually, it is a good idea to start the computations using a first-order scheme. Then, when a converged solution has been obtained, the user can continue the calculations with a second-order scheme. 

AG277).Tlieir deprotoiiatioii with strong bases led to free deeply red-colored azapeiitatriafulvaleiies which decomposed quickly (Scheme 10). 

In order to increase the accuracy of the approximation to the convective term, not only the nearest-neighbor nodes, but also more distant nodes can be included in the sum appearing in Eq. (37). An example of such a higher order differencing scheme is the QUICK scheme, which was introduced by Leonard [82]. Within the QUICK scheme, an interpolation parabola is fitted through two downstream and one upstream nodes in order to determine O on the control volume face. The un-... 

The QUICK scheme has a truncation error of order h. However, similarly as in the case of the central differencing scheme, at high flow velocities some of the coupling coefficients of Eq. (37) become negative. 

Figure 2.11 Step-function concentration profile and typical numerical results obtained with the upwind (left) and the QUICK scheme (right).

Cydizations proceeding by intramolecular Sn2 reactions are usually irreversible, and will, therefore, not necessarily yield the thermodynamically most stable product but that which is formed most quickly. Scheme 9.13 depicts a cyclization in which the outcome depends on the configuration of the starting alkene. Interestingly, the cis isomer gives mainly rise to a strained, eight-membered ether even though a path to the less strained tetrahydropyran is, in principle, accessible. 

There are couple of measures that can taken in order to minimize ND [37]. Higher order discretization schemes such as the QUICK scheme reduce the numerical errors. Furthermore, ND depends strongly on the relative orientation of flow velocity and grid cells. ND can be minimized by choosing grid cells with edges parallel to the local flow velocity. 

With the catalytic system secondaiy alcohols are still oxidized quickly (Scheme 2) and in excellent yield with 2 equiv. of peracetic acid at 0 C. Primary alcohols are also reported to be oxidized to aldehydes in good yields, but details have not been given. 

FIGURE 6.4 Quadratic profiles used in QUICK scheme. 

Hayase, T, Humphrey, J.A.C. and Grief, R. (1992), A constantly formulated QUICK scheme for fast and stable convergence using finite volume iterative calculation procedures, J. Computat. Phys., 98, 108-118. 

Quadratic Upwind Interpolation for Convective Kinematics (QUICK) Scheme... 

The quadratic upstream interpolation for convective kinetics (QUICK) scheme of Leonard [106] uses a three-point upstream-weighted quadratic interpolation for the cell face values. In the third order QUICK scheme the variable profile between P and E is thus approximated by a parabola using three node values. At location e on a uniform Cartesian grids, tpe is approximated as ... 

The main advantage of this scheme is that it is less numerically diffusive than the UDS. However, the scheme is dispersive and a severe disadvantage associated with the QUICK scheme is that it it does not satisfy the boundedness criteria unconditionally. The QUICK scheme is therefore conditionally stable. The explicit QUICK scheme is unstable in the absence of diffusion. 

The implicit QUICK scheme may become unstable due to the possible appearance of negative coefficients in the coefficient matrix, hence a reallocation of the terms in the coefficient matrix is required to alleviate these stability problems. To retain positive coefficients in the coefficient matrix the troublesome negative terms (possibly amongst others) are placed in the source term. One approach is to put the upwind coefficients into the coefficient matrix and the deviation between the upwind and QUICK coefficient values into the source term. The corresponding coefficients in (12.88) are defined as ... 

It can be shown that the leading truncation error associated with the QUICK scheme is proportional to the grid spacing in the power 3. It is noted that, for cases where (v n)g > 0, for a general GCV on a uniform grid the QUICK scheme determines the value of ipe at the grid cell face e by the first approximation in (12.104). 

The SHARP and SMART schemes are monotone versions of the QUICK scheme. 

For the mechanism of the curing of the epoxides with the imidazoles, as seen from previous studies and the experimental data obtained by Jisova [103], the reaction scheme shown in Schemes 6-9 is suggested. The initiation occurs with an induction period. The 1 1 (resp., 1 2) adducts of the imidazole and the epoxide are formed quickly (Scheme 6). In the presence of the excess epoxides, growth centers appear after the dissociation or the rearrangement of the formed adduct (Scheme 7). The addition of a further molecule of epoxide to the growth center starts the propagation of polymer (Scheme 8). 

In the computation of the advection term in Eq. (1.33), a high-resolution scheme should be employed to ensure bounded but also accurate enough solutions as discussed in [4]. In this work, the CUBISTA [1] scheme is employed, which uses QUICK scheme as the core scheme and fulfills the Total Variation Diminishing (TVD) constraint. 

The nucleophilic addition of n-BuLi to aldimine (14) afforded amine (15), (16) in diastereoselective manner. Interestingly, the effect of solvent system for the reactivity and diastereoselectivity is drastic. Although the reaction in TMEDA/toluene proceeded with low diastereoselectivity, in Et20 highly diastereoselective reaction was promoted quickly (Scheme 3.33) [68]. 

In addition to the central differencing and upwind differencing schemes, which are first-order schemes, another popular finite difference scheme is the QUICK scheme, a second-order upwind differencing scheme. Higher order means that more node points are involved when estimating the values of the dependent variables and their derivatives for formulating the finite difference equations. 

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