Operator splitting techniques

The explicit operator splitting techniques based on sequential one-dimensional updates may contain large splitting errors. To reduce the splitting errors... 

Here p is the local density of the mixture and T is the temperature the mixture would attain in a state of equilibrium with the local value of density and internal energy. For the calculation, the Random Choice Method is used, adapted by an operator splitting technique to account for relaxation [11,12]. The condensation discontinuity is not a priori included in the algorithm. The numerical results are given in Fig. 3. The formation of a... 

There are potential technical problems in simulating steeply repulsive potential fluids by MD, and in response a new molecular dynamics algorithm capable of integrating the equations of motion of impulsive-continuous potential systems (e.g. such as a hard sphere core combined with an r tail) has been derived using operator splitting techniques by Houndonougbo et... 

Sportisse, B. An analysis of operator splitting techniques in the stiff case. J. Comput. Phys. 161,... 

In addition to the direct solution of a set of matrix equations, several approximate solution methods involve what can be called operator splitting techniques. To begin this discussion, consider again the simple Poisson equation in the form ... 

One more example of a classical iterative method of solving a set of matrix equations will now be considered. For this approach take a look back at the operator splitting technique as presented in Eq. (12.67) for Poisson s equation. This technique splits the solution into two essentially one dimensional problems in terms of iterating on the matrix equations. When finite differences are applied to... 

The alternating direction methods, reviewed by Douglas and Gunn (1964), solve this by an operator-splitting technique as follows... 

This paper presents a mathematical model and numerical analysis of momentum transport and heat transfer of polymer melt flow in a standard cooling extruder. The finite element method is used to solve the three-dimensional Navier-Stokes equations based on a moving barrel formulation a semi-Lagrangian approach based on an operator-splitting technique is used to solve the heat transfer advection-diffusion equation. A periodic boundary condition is applied to model fully developed flow. The effects of polymer properties on melt flow behavior, and the additional effects of considering heat transfer, are presented. 

An alternative relax-and-drive procedure can be based on a strictly unitary treatment where the advance from Iq to t is done with a norm-conserving propagation such as provided by the split-operator propagation technique.(49, 50) This however is more laborious, and although it conserves the norm of the density matrix, it is not necessarily more accurate because of possible inaccuracies in the individual (complex) density matrix elements. It can however be used to advantage when the dimension of the density matrix is small and exponentiation of matrices can be easily done.(51, 52)... 

A reactive transport model in a more general sense treats a multicomponent system in which a number of equilibrium and perhaps kinetic reactions occur at the same time. This problem requires more specialized solution techniques, a variety of which have been proposed and implemented (e.g., Yeh and Tripathi, 1989 Steefel and MacQuarrie, 1996). Of the techniques, the operator splitting method is best known and most commonly used. 

Using the operator equivalent technique of Stevens (32), and following Elliott and Stevens (33), one may also express the splittings in terms of the better known A parameters (c.f. also Abragam and Bleaney (34)), which are in turn simply related to the Bk parameters which occur in the tensor operator treatment of crystal field splittings. Using the relationship given by Wyboume (35), the crystal field and Bk parameters are found to be connected by the expressions... 

Operator splitting methods require less computational resources but demand more thought. In general, stability is not guaranteed. However, operator splitting encourages modular models and allows the use of the best available numerical technique for each module. Thus advection, kinetics, cloud processes, and aerosol processes each reside in different modules. Techniques used for these subproblems will be outlined subsequently. For applications in other problems, the reader is referred to Oran and Boris (1987). 

The Hierarchical Approach developed in this chapter incorporates a knowledge-based procedure for the synthesis of separations. This consists of dividing the separation section in three subsystems gas vapour, liquid and solid separations. Each subsystem is further managed by selectors, which makes use of unit operations. Split sequencing is based mainly on heuristics, although may include algorithmic or optimisation techniques. This chapter describe in more detail the synthesis of distillation trains for zeotropic distillations, the non-ideal case being left for the Chapter 9. 

The use of high-resolution schemes in interfacial flow calculations is promising, but existing schemes are still too diffusive for the capmring of sharp interfaces. Implementation in multiple dimensions is also restricted to the technique of operator splitting. A generalization for implementation on unstructured meshes is still required. 

The fuel compacts were in the form of 2.01-in. cubes. Each cube was spray-painted with one-mil-thick coat of aluminum paint and 11-mil-thick rubberized plastic. The approach-to-critical technique was used to assemble the critical arrays utilizing a remotely operated split-table device. A summary of the experimental data is given in Table I. Also given are the criticality data for an equivalent Pu/water mixture fully reflected by water and having an H/Pu atomic ratio of 15 and a plutonium density of 1.52 g/cm. The conversion factors were obtained from a 12-group diffusion-theory calculation. 

The bottles were placed in arrays on a remotely operated split-table device, using neutron multiplication techniques to estimate the number of bottles for criticality. Extrapolations were also made on spacing to determine the critical spacing for fixed-number square arrays of the bottles. Arrays were single tier, except in one experiment where a double-tier array was used. 

Combescure, D. Pegon, P. 1997. a-operator splitting time integration technique for pseudodynamic testing, Error propagation analysis. Soil Dynamics and Earthquake Engineering. Vol. 16, pp. 427-443. 

Since the early 1960s, there has been much discussion in the literature of syringe technique during the injection process. Most of this related to facilitating rapid and quantitative sample transfer into the inlet, despite slow, nonreproducible manual operation. These techniques, including cold needle, hot needle, solvent flush, and inclusion of air along with the sample, are not discussed in detail here, as for most hot injection methods (split and splitless), a fast autoinjector, readily available for all gas chromatographs and used as the standard in most industries, will provide excellent quantitative injections. For more information on manual injection techniques the reader is referred to Reference 3. 

This procedure is then repeated after each time step. Comparison with Eq. (2) shows that the result is the velocity Verlet integrator and we have thus derived it from a split-operator technique which is not the way that it was originally derived. A simple interchange of the Ly and L2 operators yields an entirely equivalent integrator. 

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