Variable time-step approach

In the variable time-step approach, the system moves in event space, thus simulating the elementary kinetic processes event-by-event whereby the time is updated in variable time incrementsl . At any instant in time, ti, the rates for all possible events are... 

A second drawback of the fixed-time approach is that it is mathematically not exact. The accuracy of the simulation is governed by the choice of the time step used. Only in the limit of an infinitesimal time step does the method becomes mathematically exact. For simulations performed at very small time steps, accuracy is not an issue. Although the fixed time algorithm has proven to be fairly effective for certain gas-phase reaction systemsl " , nearly all of the published studies on surfaces use what is known as the variable time-step approachl . 

Since 5 is a function of all the intermediate coordinates, a large scale optimization problem is to be expected. For illustration purposes consider a molecular system of 100 degrees of freedom. To account for 1000 time points we need to optimize 5 as a function of 100,000 independent variables ( ). As a result, the use of a large time step is not only a computational benefit but is also a necessity for the proposed approach. The use of a small time step to obtain a trajectory with accuracy comparable to that of Molecular Dynamics is not practical for systems with more than a few degrees of freedom. Fbr small time steps, ordinary solution of classical trajectories is the method of choice. 

The study of isotopes makes it necessary to introduce a further refinement in the general method of solution. I have been using a test of the relative increment to adjust the time step. The relative increment is the change in a dependent variable divided by the value of that variable. This is not a useful test, however, when the value of the variable approaches zero, because the test requires progressively smaller time steps. None of the variables I considered in previous chapters has approached zero, and so there has been no problem with this test. But carbon isotope ratios of seawater have delta values near zero, and a problem may occur when calculating these values. I have modified subroutine CHECKSTEP to permit a flexible response to this situation. 

Any of the global Newton methods can be converted to a relaxation form in Ketchum s method by making both the temperatures and the liquid compositions time dependent and by having the time step increase as the solution is approached. The relaxation technique should be applied to difflcult-to-solve systems and the method of Naphtali and Sandholm (42) is best-suited for nonideal mixtures since both the liquid and vapor compositions are included in the independent variables. Drew and Franks (65) presented a Naphtali-Sandholm method for the dynamic simulation of a reactive distillation column but also stated that this method could be used for finding a steady-state solution. 

How one forms the approximations for the ODEs is crucial to the performance of this approach. Gear [22] and many others since showed how implicit methods convert the ODEs so that the solution method is stable and can therefore be used to solve stiff sets of equations. Implicit methods give algebraic equations that generally must be solved iteratively at each time step, as they usually involve the variables at time step k + 1 nonlinearly. 

Obviously, the above algorithms are not suitable when transients of the finer scale model are involved (Raimondeau and Vlachos, 2000), as, for example, during startup, shut down, or at a short time after perturbations in macroscopic variables have occurred. The third coupling algorithm attempts fully dynamic, simultaneous solution of the two models where one passes information back and forth at each time step. This method is computationally more intensive, since it involves continuous calls of the microscopic code but eliminates the need for a priori development of accurate surfaces. As a result, it does not suffer from the problem of accuracy as this is taken care of on-the-fly. In dynamic simulation, one could take advantage of the fast relaxation of a finer (microscopic) model. What the separation of time scales between finer and coarser scale models implies is that in each (macroscopic) time step of the coarse model, one could solve the fine scale model for short (microscopic) time intervals only and pass the information into the coarse model. These ideas have been discussed for model systems in Gear and Kevrekidis (2003), Vanden-Eijnden (2003), and Weinan et al. (2003) but have not been implemented yet in realistic MC simulations. The term projective method was introduced for a specific implementation of this approach (Gear and Kevrekidis, 2003). 

The on-line implementation of environmental processes in the GEM model allows running in global uniform, global variable, and limited area configurations, allowing for multiscale chemical weather forecasting (CWF) modelling. This approach provides access to all required dynamics and physics fields for chemistry at every time step. The on-line implementation of chemistry and aerosol processes... 

For TIDEP the parallelization can be pushed at a fine grain level by focusing on the time propagation routine (AV) that in our code is based on a Discrete variable representation (DVR) approach [24, 40]. The routine propagates the system wavepacket by repeating at eacJi time step the following stream of matrix operations... 

Aside from trying to cope with the large number of parameters, most workers have struggled somewhat with methods for numerical solution of this tj e of problem. That also is another story all by itself Some approaches have included a variable space-step method (Eigenberger and Butt [22]) and collocation (Carey and Finlayson [23], Kam and Hughes [24]). There still seems to be a good distance to go in this area, but with the advent of powerful desktop machines, computer time seems not so precious a commodity as it was not too long ago. 

Straightforward stepwise integration of the coupled Hamiltonian and L-vN differential equations would be inefficient and possibly computationally inaccurate, because the fast quantal oscillations demand very small time steps, while the slow quasiclassical motions must be followed over long times, requiring many steps. The accumulation of round-ofif errors would lead to large inaccuracies. An alternative is to separately do some of the integrations by quadratures. An obvious approach would be to use a perturbation expansion around the initial density matrix To, but this also requires small time intervals because the density matrix relaxes rapidly for fixed quasiclassical variables. An alternative solution which works well is to make a first-order perturbation correction to the relaxing (time-dependent) density matrix. 

The additional measurement of y3 improves only slightly the prediction for y but the additional measurement of y4 has virtually no effect on the prediction. This example is useful to demonstrate the approximation used in the Bayesian time-domain approach. The random variable yi is predicted by the measurements of y2. y3 and 4, which are 0.5,1 and 1.5 periods apart from yi, respectively. It turns out that including the data points within one period is sufficient. Furthermore, in a usual situation, the sampling time step is much less than half of... 

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