Numerical scheme

It was also interesting to compare LN behavior as increases to trajectories that use nonbonded cutoffs for very large /c2, behavior of the LN trajectory begins to resemble the cutoff trajectory [88]. This observation suggests that the model itself, rather than the numerical scheme per se, is responsible for the deviations. 

We assume that A is a symmetric and positive semi-definite matrix. The case of interest is when the largest eigenvalue of A is significantly larger than the norm of the derivative of the nonlinear force f. A may be a constant matrix, or else A = A(y) is assumed to be slowly changing along solution trajectories, in which case A will be evaluated at the current averaged position in the numerical schemes below. In the standard Verlet scheme, which yields approximations y to y nAt) via... 

Numerous schemes can be devised to classify deposition processes. The scheme used herein is based on the dimensions of the depositing species, ie, atoms and molecules, softened particles, Hquid droplets, bulk quantities, or the use of a surface-modification process (1,2). Coating methods are as foHow (2) ... 

The question of stiffness then depends on the solution at the current time. Consequently nonhuear problems can be stiff during one time period and not stiff during another. While the chemical engineer may not actually calculate the eigenvalues, it is useful to know that they determine the stabihty and accuracy of the numerical scheme and the step size used. 

Long, P. E., and Pepper, D. W., A comparison of six numerical schemes for calculating the advection of atmospheric pollution, in "Proceedings of the Third Symposium on Atmospheric Turbulence, Diffusion and Air Quality." American Meteorological Societv, Boston, 1976, pp. 181-186. 

Whereas Fishbum was mainly interested in the detonative mode of explosion, Luckritz (1977) and Strehlow et al. (1979) focused on the simulation of generation and decay of blast from deflagrative gas explosions. For this purpose, they employed a similar code provided with a comparable heat-addition routine. Strehlow et al. (1979), however, realized that perfect-gas behavior, which is the basis in the numerical scheme for the solution of the gas-dynamic conservation equations, is an idealization which does not reflect realistic behavior in the large temperature range considered. 

A method is outlined by which it is possible to calculate exactly the behavior of several hundred interacting classical particles. The study of this many-body problem is carried out by an electronic computer which solves numerically the simultaneous equations of motion. The limitations of this numerical scheme are enumerated and the important steps in making the program efficient on the computer are indicated. The applicability of this method to the solution of many problems in both equilibrium and nonequilibrium statistical mechanics is discussed. 

Figures 9 and 10 compare the him thickness and pressure prohles from the present model with the results calculated by Venner et al. [42] obtained under the same conditions. The comparison in Fig. 9 is made for the cases of transverse stationary sinusoidal waviness while Fig. 10 is for longitudinal waviness. Again, the results exhibit a good similarity. The good agreements in solutions from different researchers conhrm that the numerical schemes implemented in the present model are reliable, at least for the analyses in full-hlm EHL regime.

Another numerical study of free-surface flow patterns in narrow channels was conducted by Yang et al. [185]. They considered the flow of bubbles of different size driven by body forces, for example the rising of bubbles in a narrow capillary due to buoyancy. The lattice Boltzmann method [186] was used as a numerical scheme... 

One of the drawbacks of the first iteration, however, is that computation of energy quantities, e.g. orbital and total energies, requires to evaluate the integrals occurring in Eq. 3 on the basis of the ( )il )(p)- Unfortunately, the transcendental functions in terms of which the (]>il Hp) are expressed at the end of the first iteration do not lead to closed form expressions for these integrals and a numerical procedure is therefore needed. This constitutes a barrier to carry out further iterations to improve the orbitals by approaching the HE limit. A compromise has been proposed between a fully numerical scheme and the simple first iteration approach based on the fact that at the end of each iteration the < )j(k)(p) s entail the main qualitative characteristics of the exact solution and most... 

The COBRA IIIC/MIT-1 code has an improved numerical scheme, and runs faster than the COBRA IIIC code without sacrificing accuracy. 

The flow field of the impacting droplet and its surrounding gas is simulated using a finite-volume solution of the governing equations in a 3-D Cartesian coordinate system. The level-set method is employed to simulate the movement and deformation of the free surface of the droplet during impact. The details of the hydrodynamic model and the numerical scheme are described in Sections... 

Numerical Settings of Two Simulations on a Gas Cyclone With Different Numerical Schemes a RANS-Based Simulation (Hoekstra, 2000) and a LES due to Derksen and Van den... 

Numerical scheme Finite volume (FLUENT) Lattice-Boltzmann... 

It makes sense to compare the implications (in terms of simulation times) of using FV vs. LB in simulating turbulent-flow fields in process devices. Hoekstra (2000) demonstrated the numerical implications of applying different numerical schemes in an industrial application. He compared the outcome of his RANS simulation for a gas cyclone with that of a LES carried out by Derksen and Van den Akker (2000). Table I presents a number of numerical features of the two types of simulations. 

Another comparison is due to Van Wageningen et al. (2004) who performed a similar study (in terms of the numerical scheme used) on unsteady laminar flow in a Kenics static mixer. They found that the LB code was 500-600 times faster than FLUENT in terms of simulation time per grid node per time step and that FLUENT used about 5 times more memory than LB. 

Fortunately, few of these variables are truly independent. Geochemists have developed a variety of numerical schemes to solve for equilibrium in multicomponent systems, each of which features a reduction in the number of independent variables carried through the calculation. The schemes are alike in that each solves sets of mass action and mass balance equations. They vary, however, in their choices of thermodynamic components and independent variables, and how effectively the number of independent variables has been reduced. 

The model equation for particle position, (7.27), is a stochastic differential equation (SDE). The numerical solution of SDEs is discussed in detail by Kloeden and Platen (1992).28 Using a fixed time step At, the most widely used numerical scheme for advancing the particle position is the Euler approximation ... 

Higher-order numerical schemes are also available (Kloeden and Platen 1992), but are generally complicated to apply since they involve derivatives of the coefficients. A simpler alternative is to apply a multi-step approach (Pope 1995 Jenny el al. 2001). For example, the mid-point position,30... 

Implementation of a covariance structure into this numerical scheme is described in Tarantola and Valette (1982). In essence, an a priori covariance structure is assumed for the whole set of observations and parameters, which should be tightened by iterative refinements since we are still dealing with a minimum variance estimate. 

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