Partial third-order method

Similar accuracy (0.2-0.3 eV for valence ionization energies below 20 eV) is available using propagator methods in conjunction with triple-f basis sets. These propagator methods are known in the literature as the outer valence Green s function (OVGF) and partial-third order (P3)... 

The function 7(f) can be chosen for the whole reaction time interval, or two or three subsequent temperature-time data points 7(fi-i), 7(fi), and 7(fi+i) can be approximated by polynomials of second or third order 7,(f), respectively. These polynomials will then be used in a procedure for numerical integration in each integration step i. This method has been successfully applied in a kinetic study of the partial oxidation of hydrocarbons (Skrzypek et al., 1975, Krajewski etai, 1975, 1976, 1977). 

The mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation software used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over RI1/AI2O3 (Figure 8.8) demonstrates how adequate the mechanistic model is. 

We propose a simple method for the linearization of the equations, which are established in our case, based on the virtual work principle. The kinematic relations between the interconnected bodies are represented by the recursive equations. Under the small deformation assumption, the system generalized variables used in the equations are the relative joint coordinates at the connections and the deformation modal coordinates of the flexible bodies. In the linearization process, the differentiation of the kinematic terms with respect to the generalized variables must be performed. In our method, these partial derivatives are attained through the first and second order time differentiations of the body absolute angular velocities and through the first, second, and third order time differentiations of the mass center coordinates. This is the essential idea behind our method. The partial differentiation of the mechanical terms, for example, of the inertial tensors will also be presented. We have developed specific operators to perform the time differentiations. This method makes both the theoretical formulation and the programming task relatively simple, and allows fast computation. 

We have proposed a linearization procedure for the nonlinear dynamic equations which have been established in our case based on the virtual work principle. We have shown that the partial differentiation of the kinematic terms with respect to the generalized coordinates and velocities could be fulfilled through the first and second order time differentiation of the body absolute angular velocities and through the first, second, and third order time differentiation of the mass center vectors. The advantage of our method lies in the... 

A method similar to the iterative, is the partial closure method [37], It was formulated originally as an approximated extrapolation of the iterative method at infinite number of iterations. A subsequent more general formulation has shown that it is equivalent to use a truncated Taylor expansion with respect to the nondiagonal part of T instead of T-1 in the inversion method. An interpolation of two sets of charges obtained at two consecutive levels of truncations (e.g. to the third and fourth order) accelerates the convergence rate of the power series [38], This method is no longer in use, because it has shown serious numerical problems with CPCM and IEFPCM. 

The analysis of the several different CC approaches in terms of the fifth-order energy contributions points out that within an n6 dependent scheme, i.e., LCCD to CCSD in Table I, the CCSD is much preferred since it accounts for nearly one-third of all terms and avoids potential singularities in LCCD.42 It may also be observed that it pays off to include, even partially, the triple contribution, as was done in the CCSDT-1 method.10 In this model the number of terms is nearly doubled as compared to CCSD, and, of course, this method is correct through the fourth-order energy and the second-order wave function. Also, the connected T contributions are numerically important.11-34... 

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