Numerical models stochastic difference equation

Despite the similarity to the Gauss approach to classical mechanics, there is a key difference between the classical actions described above and the corresponding action of the stochastic difference equation. The classical actions are deterministic mechanical models the SDE is a nondeterministic approach that is based on stochastic modeling of the numerical errors introduced by the finite difference formula. 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

Polystochastic models are used to characterize processes with numerous elementary states. The examples mentioned in the previous section have already shown that, in the establishment of a stochastic model, the strategy starts with identifying the random chains (Markov chains) or the systems with complete connections which provide the necessary basis for the process to evolve. The mathematical description can be made in different forms such as (i) a probability balance, (ii) by modelling the random evolution, (iii) by using models based on the stochastic differential equations, (iv) by deterministic models of the process where the parameters also come from a stochastic base because the random chains are present in the process evolution. 

Chapter 4 is devoted to the description of stochastic mathematical modelling and the methods used to solve these models such as analytical, asymptotic or numerical methods. The evolution of processes is then analyzed by using different concepts, theories and methods. The concept of Markov chains or of complete connected chains, probability balance, the similarity between the Fokker-Plank-Kolmogorov equation and the property transport equation, and the stochastic differential equation systems are presented as the basic elements of stochastic process modelling. Mathematical models of the application of continuous and discrete polystochastic processes to chemical engineering processes are discussed. They include liquid and gas flow in a column with a mobile packed bed, mechanical stirring of a liquid in a tank, solid motion in a liquid fluidized bed, species movement and transfer in a porous media. Deep bed filtration and heat exchanger dynamics are also analyzed. 

It is important to emphasize, however, that our model is different from the Langevin equation, which is a stochastic differential equation. Our model has no noise in the limit of small time steps in which the numerical errors approach zero. The noise we introduce is numerical. Once we filter the rapid oscillations, it is impossible for us to recover the tme trajectory using only the low-frequency modes. The noise in the SDE approach is introduced when we approximate a differential equation by a finite difference formula and filter out high-frequency motions. 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

The numerical methods employed to solve the transported PDF transport equation are very different from standard CFD codes. In essence, the joint PDF is represented by a large collection of notional particles. The idea is similar to the presumed multi-scalar PDF method discussed in the previous section. The principal difference is that the notional particles move in real and composition space by well defined stochastic models. Some of the salient features of transported PDF codes are listed below. 

Cij and Di represent the area-averaged concentration, the intersticial velocity and the dispersion coefficient, in channel i. It woiild be interesting to derive a similar equation for bed scale averaged variables. Unfortunately, it is impossible to derive such equation in an exact manner because diffusion and percolation processes are ruled by fundamentally different elementary mechanisms, (see e.g. Broadbent et al., 1957). Actually, the stochastic model defined by Eq.5 > describes the liquid velocity distribution and could also be used to characterize nximerically the distribution of residence times i.e., the dispersion process. Achwal et al. (1979) drew attention to a procedure using a Markov chain model which led to similar results for the velocity distribution. This model remained essentially numerical and rather cumbersome. Even if Eq.5 has a simple analytical form, its mamerical application to estimate the dispersion process is also too complex for practical purposes. 

Analytical models can be classified into deterministic and stochastic. The former formulates the relationship between the known and unknown factors in the form of equations, the solution of which often requires application of numerical methods. By following prescribed rules the same result can always be obtained from the same starting conditions and initial values of known factors. In the latter, the model contains a degree of uncertainty caused by random events or variations in the values of factors, thus leading to potentially different results even when starting from the same initial conditions. 

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