Direct numerical solution

The development of combustion theory has led to the appearance of several specialized asymptotic concepts and mathematical methods. An extremely strong temperature dependence for the reaction rate is typical of the theory. This makes direct numerical solution of the equations difficult but at the same time accurate. The basic concept of combustion theory, the idea of a flame moving at a constant velocity independent of the ignition conditions and determined solely by the properties and state of the fuel mixture, is the product of the asymptotic approach (18,19). Theoretical understanding of turbulent combustion involves combining the theory of turbulence and the kinetics of chemical reactions (19—23). 

Lenhoff, J. Chromatogr., 384, 285 (1987)] or by direct numerical solution of the conservation and rate equations. For the special case of no-axial dispersion with external mass transfer and pore diffusion, an explicit time-domain solution, useful for the case of time-periodic injections, is also available [Carta, Chem. Eng. Sci, 43, 2877 (1988)]. In most cases, however, when N > 50, use of Eq. (16-161), or (16-172) and (16-174) with N 2Np calculated from Eq. (16-181) provides an approximation sufficiently accurate for most practical purposes. 

The direct numerical solutions are obtained with ODE. The steady state values appear to be 0.1403 and 0.0656. The 90% values are... 

This could be solved by trial, using a numerical integration, to find C as a function of t. However, the graph is of a direct numerical solution of Eq (1). The degradation of the catalyst Is very rapid. In practice the catalyst will proceed from the transfer line to a reactivation zone and will be recycled. 

We will later use Eq. (13) for analyzing the experiments of SHG in ppKTP and appKTP waveguides (described in section 7). For now, we compare the analytical solution of Eq. (13) with the direct numerical solution of the differential equations in Eq. (14), which account for the temporal behavior of the interacting pulses, the nonlinear losses of the SH wave, and the possible z-dependence of the phasematching condition ... 

The FDTD approach is based on direct numerical solution of the time dependent Maxwell s curl equations. In the 2D TM case the nonzero field components are E Hy and E, the propagation is along the z direction and the transverse field variations are along x. In lossless media. Maxwell s equations... 

The determination of the ground state energy and the ground state electron density distribution of a many-electron system in a fixed external potential is a problem of major importance in chemistry and physics. For a given Hamiltonian and for specified boundary conditions, it is possible in principle to obtain directly numerical solutions of the Schrodinger equation. Even with current generations of computers, this is not feasible in practice for systems of large total number of electrons. Of course, a variety of alternative methods, such as self-consistent mean field theories, also exist. However, these are approximate. 

The term parameters of the lowest two allowed transitions of ethene calculated with different methods and different choices of computational parameters (48,51,98,105) are summarized in Table I. Included in the table are results obtained with four different basis sets. In combination with these basis sets the MCD parameters were obtained in the transition-based approach through solution of Eq. (60) by direct numerical solution (labeled Direct in Table I) and by expansion in a set of transition densities according to Eq. (72) (labeled SOS ). In some cases approximate forms of the A(1) and B(1) matrices were used (labeled Approx, see Eq. (64) and the discussion following it). MCD parameters derived from a fit to a spectrum obtained by calculation of the imaginary part of the Verdet constant are labeled as Im[V]. The parameters obtained from a fit to the spectrum obtained from the approximate form of Im[V] (see Section... 

Models for the electronic structure of polynuclear systems were also developed. Except for metals, where a free electron model of the valence electrons was used, all methods were based on a description of the electronic structure in terms of atomic orbitals. Direct numerical solutions of the Hartree-Fock equations were not feasible and the Thomas-Fermi density model gave ridiculous results. Instead, two different models were introduced. The valence bond formulation (5) followed closely the concepts of chemical bonds between atoms which predated quantum theory (and even the discovery of the electron). In this formulation certain reasonable "configurations" were constructed by drawing bonds between unpaired electrons on different atoms. A mathematical function formed from a sum of products of atomic orbitals was used to represent each configuration. The energy and electronic structure was then... 

This is a functional equation for the boundary position X and the unknown constant parameter n. Upon substituting Eq. (256) into Eq. (251) an ordinary differential equation is obtained for X(t, n), and a family of curves in the phase plane (X, X) can be obtained. For n sufficiently close to unity two functions in the phase plane can be determined which serve as upper and lower bounds for the trajectories. The choice is guided by reference to the exact solution for the limiting case of constant surface temperature. It is shown that the upper and lower bounds are quite close to the one-parameter phase plane solution, although no comparison is made with a direct numerical solution. The one-parameter solution also agrees well with experiments on the solidification of aluminum under conditions of low surface heat transfer coefficient (hi = 0.02 cm.-1). 

The single-column process (Figure 1) is similar to that of Jones et al. (1). This process is useful for bulk separations. It produces a high pressure product enriched in light components. Local equilibrium models of this process have been described by Turnock and Kadlec (2), Flores Fernandez and Kenney (3), and Hill (4). Various approaches were used including direct numerical solution of partial differential equations, use of a cell model, and use of the method of characteristics. Flores Fernandez and Kenney s work was reported to employ a cell model but no details were given. Equilibrium models predict... 

There are two other main directions for the calculation of the electrostatic interaction between the solute and a surrounding dielectric continuum for molecular-shaped cavities. Both require intensive numerical calculations and are thus slower than GB methods. The first direction is the direct numerical solution of the Poisson equation for the volume polarization P(r) at a position r of the dielectric medium ... 

The direct numerical solution of the HSCC equations (95) would be impractical because of the sharply peaked nonadiabatic coupling potentials near any of the many avoided crossings. This difficulty is often circumvented by the diabatic-by-sector technique to solve, effectively, the coupled equations (95) [95,96]. By this technique, the region of p up to some large value is divided into many sectors, labeled by i. The adiabatic functions ( 2c / ) at some fixed point p = pt in each sector are employed throughout the sector as the diabatic basis functions 0 ( c) for expansion. The expansion then takes a form (i)(p, 2C) = En p 5/2F (p) (Qc), and the coupled equations in each sector i are... 

Within the Hartree-Fock approximation, calculations on molecules have almost all used the matrix SCF method, in which the HF orbitals are expanded in terms of a finite basis set of functions. Direct numerical solution of the HF equations, routine for atoms, has, however, been thought too difficult, but McCullough has shown that, for diatomic molecules, a partial numerical integration procedure will yield very good results.102 In particular, the Heg results agree well with the usual calculations, and it is claimed that the orbitals are likely to be of more nearly uniform accuracy than in the matrix HF calculations. Extensions to larger molecules should be very interesting so far, published results are available for He, Heg, and LiH. 

The mechanisms and resulting kinetic equations are shown in Figure 4. Other mechanisms are possible as well as modifications of these—e.g., disproportion termination of chain reactions, and condensation between unlike monomers. The left sides of the equations represent the reactor operator (note that all resulting differential equations are nonlinear because of the second-order propagation and termination reactions). To this is added the complexity of considering separate equations for the thousands of separate species frequently required to define completely commercially useful polymers. Solution by direct application of classical techniques is impractical or impossible in most cases even direct numerical solution is often difficult. Simplifying assumptions or special mathematical techniques must be used (described below in the calculations of MWD). 

On the applied side of quantum chaology we find serious efforts to forge the semiclassical method into a handy tool for easy use in connection with arbitrary classically chaotic systems. Quite frankly, the current status of semiclassical methods is such that they are immensely helpful in the interpretation of quantum spectra and wave functions, but are only of limited power when it comes to accurately predicting the quantum spectrum of a classically chaotic system. In this case numerical methods geared toward a direct numerical solution of the Schrodinger equation are easier to handle, more transparent, more accurate and cheaper than any known semiclassical method. It should be the declared aim of applied semiclassics to provide methods as handy and universal as the currently employed numerical schemes to solve the spectral problem of classically chaotic quantum systems. 

In this section we summarise the properties of the approximations to tc[M] discussed in Section 4 in applications to atoms. All results presented in the following [36] are based on the direct numerical solution of Eqs. (3.25-3.29) using a nuclear potential which corresponds to a homogeneously charged sphere [69]. Only spherical, i.e. closed subshell, atoms and ions are considered. Whenever suitable we use Hg as a prototype of all high-Z atoms. 

QMC methods (type III) involve a direct numerical solution of the Schrodinger equation, subject to restrictions associated with the placement of nodes in nontrivial multielectron systems. Hence, they potentially provide an exact treatment of PJT effects, just as they provide a potentially exact treatment of all other molecular properties. However, there seems to have been very little work done in using QMC to study problems involving potential energy surfaces of radicals, possibly because of the numerical uncertainty issues associated with these calculations. Nevertheless, the potential for such applications is vast, and we encourage the QMC community to explore this challenging and important area of application. 

This entry is organized in the following paragraphs First, the advanced determination of van der Waals interaction between spherical particles is described. Second, the relevant approximate expressions and direct numerical solutions for the double-layer interaction between spherical surfaces are reviewed. Third, the experimental data obtained for AFM tips having nano-sized radii of curvature and the DLVO forces predicted by the Derjaguin approximation and improved predictions are compared. Finally, a summary of the review and recommended equations for determining the DLVO interaction force and energy between colloid and nano-sized particles is included. 

The direct numerical solutions of PB equation for spheres have been reported by a number of researchers, including. In the numerical computation, PB Eq. (13) is conveniently expressed and solved in the bispherical coordinates. Due to the rotational symmetry of the interaction along the centerline, Eq. (13) simplifies into... 

Direct numerical solution using finite-difference, finite-element, and boundary-element methods have played important roles in porous-media research. During the last decade, the lattice-Boltzmann method has emerged as a preferred method for many applications, particularly in the hydrology literature. Significant advantages include relatively simple... 

D time-dependent solution of the Navicr -Stokes equations. The main reason we do not discuss these flows here is that the analytical solution techniques that we develop have had relatively little impact on the analysis or understanding of turbulent flows. The most powerfifl theoretical tools for turbulence research and for the prediction of turbulent flows are currently direct numerical solutions (DNS) of the Navier Stokes equation, typically by use of spectral techniques for discretization. Again, the interested reader will find many texts and references to modem work on turbulent flows.2... 

In this work, we use instead a direct numerical solution to extend that analysis to include changes in the charge distribution with respect to a reference... 

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