Discretization in time

Fig. 3. Quantum solution of the test system of 3.3 for e = 1/100. computed numerically using Fourier pseudospectral methods in space and a syraplectic discretization in time. Reduced g -density f t)j dg versus t and qF Initial...

When implementing the solution, some discretization in time has to be done therefore, it may be convenient to divide the system into time intervals, and approximate (8.11) and (8.12) with the difference equations, that is,... 

The variables Pd, Pu, tu, td, P, and Y will be defined only at time points corresponding to when Pp is measured i.e., these variables will be discrete in time rather than continuous and will incorporate the subscript k to denote a specific time point corresponding to each production cycle. In addition, the operator A will be used to denote that a variable is evaluated as the deviation between its actual value and a reference value corresponding to a cyclic steady state i.e., APp k is the deviation variable for product purity for the time point k and is equal to the actual product purity minus the desired (i.e., set point) purity. It is assumed in this study that measurement of the output variable APp at the time point k occurs simultaneously with the adjustments to the input variables APu and APd which affect the product fraction produced at the time point k+1. In addition, since recycle streams are not considered in this study, it follows that if APu k differs from APu k j, the full effect of this change is manifested in a change in APp from its value for the time point k to its value for time point k+1. 

The rationale of using hybrid simulation here is that a classic diffusion-adsorption type of model, Eq. (2), can efficiently handle large distances between steps by a finite difference coarse discretization in space. As often happens in hybrid simulations, an explicit, forward discretization in time was employed. On the other hand, KMC can properly handle thermal fluctuations at the steps, i.e., provide suitable boundary conditions to the continuum model. Initial simulations were done in (1 + 1) dimensions [a pseudo-2D KMC and a ID version of Eq. (2)] and subsequently extended to (2 + 1) dimensions [a pseudo-3D KMC and a 2D version of Eq. (2)] (Schulze, 2004 Schulze et al., 2003). Again, the term pseudo is used as above to imply the SOS approximation. Speedup up to a factor of 5 was reported in comparison with KMC (Schulze, 2004), which while important, is not as dramatic, at least for the conditions studied. As pointed out by Schulze, one would expect improved speedup, as the separation between steps increases while the KMC region remains relatively fixed in size. At the same time, implementation is definitely complex because it involves swapping a microscopic KMC cell with continuum model cells as the steps move on the surface of a growing film. 

Violation of (/) can be obtained in the so-called Levy flight model [20]. The simplest instance is the discrete (in time) one-dimensional case, where the particle position x(t + 1) at the time t + 1 is obtained from x(t) as follows ... 

Markov chains or processes are named after the Russian mathematician A.A.Markov (1852-1922) who introduced the concept of chain dependence and did basic pioneering work on this class of processes [1]. A Markov process is a mathematical probabilistic model that is very useful in the study of complex systems. The essence of the model is that if the initial state of a system is known, i.e. its present state, and the probabilities to move forward to other states are also given, then it is possible to predict the future state of the system ignoring its past history. In other words, past history is immaterial for predicting the future this is the key-element in Markov chains. Distinction is made between Markov processes discrete in time and space, processes discrete in space and continuous in time and processes continuous in space and time. This book is mainly concerned with processes discrete in time and space. 

Markov chains have extensively been dealt with in refs.[2-8, 15-18, 84], mainly by mathematicians. Based on the material of these articles and books, a coherent and a short "distillate" is presented in the following. The detailed mathematics is avoided and numerous examples are presented, demonstrating the potential of the Markov-chain method. Distinction has been made with respect to processes discrete in time and space, processes discrete in space and continuous in time as well as processes continuous in space and time. 

Demonstration of the fundamentals has been performed also on the basis of examples generated from unusual sources, art and the Bible. Surprisingly, biblical stories and paintings can be nicely analyzed by applying Markov chains discrete in time and space. 

MARKOV CHAINS DISCRETE IN TIME AND SPACE 2.1-1 The conditional probability... 

It should be emphasized that the transition matrix, Eq.(2-91), applies to the time interval between two consecutive service completion where the process between the two completions is of a Markov-chain type discrete in time. The transition matrix is of a random walk type, since apart from the first row, the elements on any one diagonal are the same. The matrix indicates also that there is no restriction on the size of the queue which leads to a denumerable infinite chain. If, however, the size of the queue is limited, say N - 1 customers (including the one being served), in such a way that arriving customers who find the queue full are turned away, then the resulting Markov chain is finite with N states. Immediately after a service completion there can be at most N -1 customers in the queue, so that the imbedded Markov chain has the state space SS = [0, 1,2,. .., N - 1 customers] and the transition matrix ... 

The models discrete in space and continuous in time as well as those continuous in space and time, led many times to non-linear differential equations for which an analytical solution is extremely difficult or impossible. In order to solve the equations, simplifications, e.g. linearization of expressions and assumptions must be carried out. However, if this is not sufficient, one must apply numerical solutions. This led the author to a major conclusion that there are many advantages of using Markov chains which are discrete in time and space. The major reason is that physical models can be presented in a unified description via state vector and a one-step transition probability matrix. Additional reasons are detailed in Chapter 1. It will be shown later that this presentation coincides also with the fact that it yields the finite difference equations of the process under consideration on the basis of which the differential equations have been derived. 

Throughout this chapter it has been decided to apply Markov chains which are discrete in time and space. By this approach, reactions can be presented in a unified description via state vector and a one-step transition probability matrix. Consequently, a process is demonstrated solely by the probability of a system to occupy or not to occupy a state. In addition, complicated cases for which analytical solutions are impossible are avoided. 

In general, the numerical solution of PlDEs consists of a three step procedure. First the basic set of PlDEs is discretized in the internal coordinates and expressed as a set of partially differential equations (PDFs). The PDFs are then discretized in time and the physical space coordinates using standard PDF discretization techniques in a second step. Finally the resulting set of algebraic equations are solved by use of a suitable solver. 

Monitoring data are discrete in time and space, and interpolation and sometimes even extrapolation techniques are therefore required to achieve a continuous image of the... 

The digital controller has both input and output signals discrete in time. So far we have not studied any techniques to model such systems, which from now on we will call discrete. 

Discretize in time the continuous model of a stirred tank heater. 

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. 

A finite difference scheme for discretization in time is used at this stage. In order to reduce the set of ordinary differential equations to algebraic equations, a time weighting coefficient is introduced, that allows to use several schemes explicit, implicit or the Crank-Nicolson scheme. 

Numerical solution of the system of equations above can be obtained by discretization in time [25]. The main steps of the solution process are as follows ... 

Both Hamiltonians (5.11) and (5.21) involve the maximum velocities ajx and aX, respectively. The front velocities tend to infinity in the fast reaction limit when the diffusion approximation is considered. This means that H p) depends quadratically on ap for ap small. The two models (5.11) and (5.20) differ fundamentally with respect to propagating fronts discreteness in time leads to a finite propagation rate, while the continuous-in-time model leads to an infinite velocity of propagation in the limit of fast reaction, r -> oo. The front velocity for model (5.21) is... 

This chapter develops the control system with an adaptive minimum variance controller. The plant and the controller are simulated as discrete in time (Phillips and Nagle, 1995). The plant parameters are polynomials 5(z ) and standard deviation of white noise of... 

Curran DAS, Cross MM, Lewis BA (1980) Solution of parabolic differential equations by the boundary element method using discretization in time. Appl Math Model 4 398-400 Dai SC, Qi F, Tanner RI (2006) Strain and strain-rate formulation for flow-induced crystallization. Polym Eng Sci 46 659-669... 

Numerical integration (sometimes referred to as solving or simulation) of differential equations, ordinary or partial, involves using a computer to obtain an approximate and discrete (in time and/or space) solution. In chemical kinetics, these differential equations are typically the rate laws that describe the time evolution of the system. One obtains results for the mean concentrations, without any information about the (typically very small) fluctuations that are inevitably present. Continuation and sensitivity analysis techniques enable one to extrapolate from a numerically obtained solution at one set of parameters (e.g., rate constants or initial concentrations) to the behavior of the system at other parameter values, without having to carry out a full numerical integration each time the parameters are changed. Other approaches, sometimes referred to collectively as stochastic methods (Gardiner, 1990), can provide data about fluctuations, but these require considerably more computational labor and are often impractical for models that include more than a few variables. 

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