Temporal discretization

The selection of a time increment dependent on parameter a (i.e. carrying out Taylor series expansion at a level between successive time steps of n and n+Y) enhances the flexibility of the temporal discretizations by allowing the introduction of various amounts of smoothing in different problems. The first-order time derivatives are found from the governing equations as... 

When transient problems are considered, the time derivative appearing in Eq. (32) also has to be approximated numerically. Thus, besides a spatial discretization, which has been discussed in the previous paragraphs, transient problems require a temporal discretization. Similar to the discretization of the convective terms, the temporal discretization has a major influence on the accuracy of the numerical results and numerical stability. When Eq. (32) is integrated over the control volumes and source terms are neglected, an equation of the following form results ... 

More basically, LB with its collision rules is intrinsically simpler than most FV schemes, since the LB equation is a fully explicit first-order discretized scheme (though second-order accurate in space and time), while temporal discretization in FV often exploits the Crank-Nicolson or some other mixed (i.e., implicit) scheme (see, e.g., Patankar, 1980) and the numerical accuracy in FV provided by first-order approximations is usually insufficient (Abbott and Basco, 1989). Note that fully explicit means that the value of any variable at a particular moment in time is calculated from the values of variables at the previous moment in time only this calculation is much simpler than that with any other implicit scheme. 

The most successful NQD preparations with respect to nanocrystal quality and monodispersity entail pyrolysis of metal-organic precursors in hot coordinating solvents (120 360 °C). Understood in terms of La Mer and Dinegar s studies of colloidal particle nucleation and growth, these preparative routes involve a temporally discrete nucleation event followed by relatively rapid growth from solution-phase monomers and finally slower growth by Ostwald... 

Despite its great potential, in the near future CFD will not completely replace experimental work or standard approaches currently used by the chemical engineering community. In this connection it is even not sure that CFD is guaranteed to succeed or even be an approach that will lead to improved results in comparison with standard approaches. For single-phase turbulent flows and especially for multiphase flows, it is imperative that the results of CFD analysis somehow be compared with experimental data in order to assess the validity of the physical models and the computational algorithms. In this connection we should mention that only computational results that possess invariance with respect to spatial and temporal discretization should be confronted with experimental data. A CFD model usually gives very detailed information on the temporal and spatial variation of many key quantities (i.e., velocity components, phase volume fractions, temperatures, species concentrations, turbulence parameters), which leads to in-... 

Mass balance of solid Mass balance of water Mass balance of air Momentum balance for the medium Internal energy balance for the medium The resulting system of Partial Differential Equations is solved numerically. Finite element method is used for the spatial discretization while finite differences are used for the temporal discretization. The discretization in time is linear and the implicit scheme uses two intermediate points, t and t between the initial 1 and final t limes. Finally, since the problems are nonlinear, the Newton-Raphson method has been adopted following an iterative scheme. 

In 1993, another milestone in the preparation of II-VI semiconductor nanocrystals was the study of Murray, Norris and Bawendi [12], whose synthesis was based on the pyrolysis of organometallic reagents such as dimethyl cadmium and trioctylphosphine selenide, after injection into a hot coordinating solvent. This approach provided a temporally discrete nucleation and permitted a controlled growth of the nanocrystals. The evolution of the absorption spectra of a series of CdSe crystallites, ranging in size from about 1.2 to 11.5 nm, is shown in Figure 3.11. 

Adaptive computations of nonlinear systems of reaction-diffusion equations play an increasingly important role in dynamical process simulation. The efficient adaptation of the spatial and temporal discretization is often the only way to get relevant solutions of the underlying mathematical models. The corresponding methods are essentially based on a posteriori estimates of the discretization errors. Once these errors have been computed, we are able to control time and space grids with respect to required tolerances and necessary computational work. Furthermore, the permanent assessment of the solution process allows us to clearly distinguish between numerical and modelling errors - a fact which becomes more and more important. 

The Langevin equation is discretized temporally by a set of equally spaced time intervals. At predetermined times, the ion dynamics is frozen, and the spatial distribution of the force is calculated from the vector sum of all its components, including both the long-range and the short-range contributions. The components of the force are then kept constant, while the dynamics resumes under the effect of the updated field distribution. Self-consistency between the force field and the ionic motion in the phase space is obtained by iterating this procedure for a desired amount of simulation time. The choice of the spatial and temporal discretization schemes plays a crucial role in computational performance and model accuracy. 

The numerical solution of the governing equations (1) proceeds with the consideration of their spatial and temporal discretizations. The spatial discretization of interest here... 

Verification Verification is primarily a mathematical issue [47]. The major sources of errors in the numerical solution have been listed in ref [48] insufficient spatial and temporal discretization convergence insufficient convergence of an iterative procedure computer round-off computer programming errors. Errors of the latter type are the most difficult to detect and fix when the code executes without an obvious crash, yielding moderately incorrect results [48]. The study reported in ref. [49] revealed a surprisingly large number of such faults in the tested scientific codes (in total, over a hundred both commercial and research codes regularly used by their intended users). 

Courant condition when using an explicit time integration method (e.g., central difference method) for such small elements leads to small time increments and a huge computational cost. To resolve this problem, the Newmark-/ method (jS = 1/4, 5 = 1/2) is used for the time integration. Note that an explicit scheme is selected to compute the response robustly an implicit scheme sometimes does not reach an accurate solution because of complex nonlinear constitutive relation. Thus, to verify the convergence of a solution, attention must be paid to the spatial and temporal discretization. 

The dynamic equation of motion can be solved by numerical integration. The numerical integration calls for temporal discretization (i.e., system of coupled equations is discretized temporally), hence the term solution in time domain. The dynamic equation is solved by a time-stepping scheme. Examples of time-stepping schemes include Wilson s theta (0) algorithm (Clough and Penzien 1993) and several variations of the Newmark s p algorithms (Newmark 1959). 

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