Time discretization

Needless to mention, the exact capturing of time presents further challenges in the analysis. Fundamentally, a decision has to be made on how the time horizon has to be represented. Early methods relied on even discretization of the time horizon (Kondili et al., 1993), although there are still methods published to date that still employ this concept. The first drawback of even time discretization is that it inherently results in a very large number of binary variables, particularly when the granularity of the problem is too small compared to the time horizon of interest. The second drawback is that accurate representation of time might necessitate even smaller time intervals with more binary variables. Even discretization of time is depicted in Fig. 1.8a. 

The choice of time discretization is determined by preference and the complexity of the equation to be solved. Explicit differences are simple and do not require iteration, but they can require small time steps for an accurate solution. Imphcit and fully implicit differences allow for accurate solutions at significantly larger time steps but require additional computations through the iterations that are normally required. 

Once the PDE has been semi-discretized (i.e., discretize the spatial derivatives but not the timelike derivatives) to form a system of ODEs, the ODEs can be solved by high-level software packages. In the standard form there are many such packages available, with relatively fewer for DAEs (see Section 15.3.3). In the method of lines, the spatial differencing must be done by the user, while time discretization and error control is handled by the ODE software. Overall, the effort to develop a new simulation is reduced, since a good deal of existing high-level software can be used. 

Numerical models such as the one by Moridis et al. (2005) are more sophisticated, and are subject to space- and time-discretization errors so numerical models must be compared to some standard, such as a physical measurement, or perhaps an analytical model. 

In the case of dynamic (unsteady) problems, even after the space discretization, we still have to solve a set of ordinary differential equations in time. Therefore, the second step is to discretize the temporal continuum. This is usually done by a finite difference approximation with the same properties of a FDM in space. Depending on the instant in which the information is taken, the time-discretization leads to ... 

Now, the effective linear response function h(t) can be identified with g(t) as defined in Eqs. (25) and (29) h(t) = g(t). The primary sample response is the heterodyne diffraction efficiencyy (t) = Chet(t)- The instantaneous contribution of the temperature grating to the diffraction efficiency is expressed by the 5-function in g(t) [Eq. (25)]. After the sample, an unavoidable noise term e(t) is added. The continuous yff) is sampled by integrating with an ideal detector over time intervals At to finally obtain the time-discrete sequence y[n]. 

Hence, the TDFRS experiment converts an ideal time-discrete excitation x[n] into a time discrete signaly[n], and the task is to extract the response h(t) =g(t) from the measuredy[n] and the known x[n] according to Eq. (61). 

The most straightforward way to compute a path integral numerically is to make time discrete for the propagator (3.1) ... 

As an alternative to making time discrete, we mention the methods based on Fourier expansion of the path with subsequent integration over Fourier coefficients. These methods are mostly applied to calculate statistical properties at finite temperatures (see, e.g., Doll and Freeman [1984], Doll et al. [1985], Topper and Truhlar [1992], and Topper et al. [1992]). 

Vol. 529 W. Krabs, S. W. Pickl, Analysis, Controllability and Optimization of Time-Discrete Systems and Dynamical Games. XII, 187 pages. 2003. 

The most common methodology when solving transient problems using the finite element method, is to perform the usual Garlerkin weighted residual formulation on the spatial derivatives, body forces and time derivative terms, and then using a finite difference scheme to approximate the time derivative. The development, techniques and limitations that we introduced in Chapter 8 will apply here. The time discretization, explicit and implicit methods, stability, numerical diffusion etc., have all been discussed in detail in that chapter. For a general partial differential equation, we can write... 

Equation (11) represents the time-discrete dynamic equivalent of the steady-state balance equations (2). The dynamic balance equations (11) present some characteristic properties of the sampled-data input and output relationship, that are not present in the corresponding steady-state equations 1) There are as many equations as the number of outputs 2) Each equation contain only one output 3) Each equation contain, except for special cases, all the inputs variables. 

Elden L (1982) Time discretization in the backward solution of parabolic equations. II. Math Comput 39 69-84... 

In the simulation of the evolution of the dye concentration the radiative and kinetic problems cannot be decoupled, as was the case for nonabsorbing contaminants. Therefore, some kind of numerical procedure based on time discretization has to be applied for the solution of Equation (10). One possibility is to apply an Euler-t)q5e method that consists in the following steps (Villafan-Vidales et al., 2007) ... 

The SDE and transport equation can be used with the same univocity conditions. For simple univocity conditions and functions such as Di-a(Fa), the transport equations have analytical solutions. Comparison with the numerical solutions of stochastic models allows one to verify whether the stochastic model works properly. The numerical solution of SDE is carried out by space and time discretization into space subdivisions called bins. In the bins j of the space division i, the dimensionless concentration of the property (F = Fa/Faq) takes the Fj value. Taking into consideration these previous statements allows one to write the numerical version of relation (4.118) ... 

The method of characteristics, the distance method of lines (continuous-time discrete-space), and the time method of lines (continuous-space discrete-time) were used to solve the solids stream partial differential equations. Numerical stiffness was not considered a problem for the method of characteristics and time method of lines calculations. For the distance method of lines, a possible numerical stiffness problem was solved by using a simple sifting procedure. A variable-step fifth-order Runge-Kutta-Fehlberg method was used to integrate the differential equations for both the solids and the gas streams. 

After phase inversion, the dispersed phase is the rubber phase. This is the first time discrete rubber particles are present in the reaction mixture. The particle size in the final product is an important parameter to optimize the physical properties. To be successful in the manufacturing of mass ABS, it is necessary to understand and control which parameters can be used to control the final rubber particle size. 

In practice, the differential equation (Equation [1]) is solved by transforming it into a difference equation employing a finite time step. The resulting time discretization error is dealt with in a number of publications.42-46 Simplification of Equation [3] is often required to expedite the calculation and therefore allow generation of longer trajectories. One approximation (for nonpolar species) is to write... 

The complete mass balance model for the described network can be found in [1]. Since the model in [1] has tactical purposes, an LP approach with a weekly time discretization was adopted in that study. 

The zero-temperature string method was primarily formulated as a time-continuous evolution of a curve F t) rather than the time-discrete evolution as explained here, but this difference is not essential. 

The easiest nontrivial example is a time-discrete Markov chain on a discrete state space. For example, take the chain with state space S — 1, 2,3,4 and one-step transition probabilities as illustrated in Fig. 1. 

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