Difference implicit

Note that, in loeal eoordinates. Step 2 is equivalent to integrating the equations (13). Thus, Step 2 can either be performed in loeal or in eartesian coordinates. We consider two different implicit methods for this purpose, namely, the midpoint method and the energy conserving method (6) which, in this example, coineides with the method (7) (because the V term appearing in (6) and (7) for q = qi — q2 is quadratie here). These methods are applied to the formulation in cartesian and in local coordinates and the properties of the resulting propagation maps are discussed next. 

In Section III we described an approximation to the nonpolar free energy contribution based on the concept of the solvent-accessible surface area (SASA) [see Eq. (15)]. In the SASA/PB implicit solvent model, the nonpolar free energy contribution is complemented by a macroscopic continuum electrostatic calculation based on the PB equation, thus yielding an approximation to the total free energy, AVP = A different implicit... 

Figure8-1 Space-time grid for the one-dimensional diffusion equation, evidencing the explicit forward-difference, implicit backward-difference and C rank-Nicholson discretization schemes.

An improved simulation might therefore be obtained by using an estimate of the average concentration value during the period Af. This is done in the implicit method, which considers the previous data point as well as the next, yet to be determined point in the computation. In fact, there are many different implicit methods. Here we only illustrate the simplest of them, which assumes that all variables change linearly over a sufficiently small interval A f. 

Mass balance equations of pesticide fate and transport are developed for the surface and subsurface zones in PRZM. In the surface zone, avenues of loss include soluble loss in runoff, percolation to the next zone, sortoed loss in erosion, and decay in both phases. In the subsurface zones, losses include plant uptake and percolation in the soluble phase, and decay in both phases. A backward difference, Implicit numerical scheme is used to solve the partial differential solute transport equations, with a time step of one day and a spatial increment specified by the user. 

After a transition from a different refinement program to SHELXL, the 5-values are different from what you had before. This can be caused by differences in the model, for example the use of different bulk solvent corrections or by different sets of reflections used for the calculation of the 5-values. In particular, the use of different sigma-cutoffs (e.g. the exclusion of reflections that have an an litude of less than say 3 times their standard deviation, f < 3a (F) ) will result in different statistics. Checking the exact number of reflections will reveal this problem and others such as different implicit resolution cut-offs. 

For power law fluids, Tomlta (1959) extended his laminar flow model (discussed in Section 5.2.1.3) to turbulent flows in smooth pipes by applying Prandtl s mixing length concept, and developed a different implicit equation ... 

In the work presented here, a slightly different two-parameter transient model has been used. Instead of specifying a center frequency b and the bandwidth parameter a of the amplitude function A(t) = 6 , a simple band pass signal with lower and upper cut off frequencies and fup was employed. This implicitly defined a center frequency / and amplitude function A t). An example of a transient prototype both in the time and frequency domain is found in Figure 1. 

In an ambitious study, the AIMS method was used to calculate the absorption and resonance Raman spectra of ethylene [221]. In this, sets starting with 10 functions were calculated. To cope with the huge resources required for these calculations the code was parallelized. The spectra, obtained from the autocorrelation function, compare well with the experimental ones. It was also found that the non-adiabatic processes described above do not influence the spectra, as their profiles are formed in the time before the packet reaches the intersection, that is, the observed dynamic is dominated by the torsional motion. Calculations using the Condon approximation were also compared to calculations implicitly including the transition dipole, and little difference was seen. 

A reasonable approach for achieving long timesteps is to use implicit schemes [38]. These methods are designed specifically for problems with disparate timescales where explicit methods do not usually perform well, such as chemical reactions [39]. The integration formulas of implicit methods are designed to increase the range of stability for the difference equation. The experience with implicit methods in the context of biomolecular dynamics has not been extensive and rather disappointing (e.g., [40, 41]), for reasons discussed below. 

The implicit-midpoint (IM) scheme differs from IE above in that it is symmetric and symplectic. It is also special in the sense that the transformation matrix for the model linear problem is unitary, partitioning kinetic and potential-energy components identically. Like IE, IM is also A-stable. IM is (herefore a more reasonable candidate for integration of conservative systems, and several researchers have explored such applications [58, 59, 60, 61]. 

The many approaches to the challenging timestep problem in biomolecular dynamics have achieved success with similar final schemes. However, the individual routes taken to produce these methods — via implicit integration, harmonic approximation, other separating frameworks, and/or force splitting into frequency classes — have been quite different. Each path has encountered different problems along the way which only increased our understanding of the numerical, computational, and accuracy issues involved. This contribution reported on our experiences in this quest. LN has its roots in LIN, which... 

In the case of unmixed vapors between the plates, the equations, being implicit in Ey, have also been solved numerically (112). The results depend on the arrangement of the downcomers and are not too different numerically from equation 93. In reaHty, however, the Hquid is neither completely backmixed nor can the tray be considered as a plug-flow device. 

Helium-3 [14762-55-1], He, has been known as a stable isotope since the middle 1930s and it was suspected that its properties were markedly different from the common isotope, helium-4. The development of nuclear fusion devices in the 1950s yielded workable quantities of pure helium-3 as a decay product from the large tritium inventory implicit in maintaining an arsenal of fusion weapons (see Deuterium AND TRITIUM) Helium-3 is one of the very few stable materials where the only practical source is nuclear transmutation. The chronology of the isolation of the other stable isotopes of the hehum-group gases has been summarized (4). 

The primary process of SiH decomposition is electron impact which produces a large number of different neutral and ionic species as shown in Table 1. The density of S1H2 and SiH neutral species produced has been found to be much larger than the density of the ions. For example, mass spectrometric data for silane discharges indicate that the density of ionic species is lower by 10 compared with the density of neutral species. Further, mass spectrometer signals of ionic species, such as SiH SiH 25 SiH", SiH", and Si2H , increase by more than two orders of magnitude as the r-f power is increased, eg, from 2 to 20 W. A rapid rise in the population of ions, with power, implicitly means an increase in electron density. 

Implicit Methods By using different interpolation formulas involving y, it is possible to cferive imphcit integration methods. Implicit methods result in a nonhnear equation to be solved for y so that iterative methods must be used. The backward Euler method is a first-order method. 

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

Implicit methods can also be used. Write a finite difference form for the time derivative and average the right-hand sides, evaluated at the old and new time. 

Cullinan presented an extension of Cussler s cluster diffusion the-oiy. His method accurately accounts for composition and temperature dependence of diffusivity. It is novel in that it contains no adjustable constants, and it relates transport properties and solution thermodynamics. This equation has been tested for six very different mixtures by Rollins and Knaebel, and it was found to agree remarkably well with data for most conditions, considering the absence of adjustable parameters. In the dilute region (of either A or B), there are systematic errors probably caused by the breakdown of certain implicit assumptions (that nevertheless appear to be generally vahd at higher concentrations). 

D Cregut, J-P Liautard, L Chiche. Homology modeling of annexm I Implicit solvation improves side-chain prediction and combination of evaluation criteria allows recognition of different types of conformational eiTor. Protein Eng 7 1333-1344, 1994. 

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