Selectivity stiff differential equations

Stiff differential equations Differential equations with widely varying rate constants. Like neural networks, their solution depends upon careful selection of step sizes. 

The numerical solution of the equation was achieved by using a library routine which selected the order and step size automatically and was designed for stiff differential equations (IMSL-DGEAR). The routine was applied in the following iterative scheme ... 

The successful numerical solution of differential equations requires attention to two issues associated with error control through time step selection. One is accuracy and the other is stability. Accuracy requires a time step that is sufficiently small so that the numerical solution is close to the true solution. Numerical methods usually measure the accuracy in terms of the local truncation error, which depends on the details of the method and the time step. Stability requires a time step that is sufficiently small that numerical errors are damped, and not amplified. A problem is called stiff when the time step required to maintain stability is much smaller than that that would be required to deliver accuracy, if stability were not an issue. Generally speaking, implicit methods have very much better stability properties than explicit methods, and thus are much better suited to solving stiff problems. Since most chemical kinetic problems are stiff, implicit methods are usually the method of choice. 

Vode, solves stiff systems of ordinary differential equations (ODE) using backward differentiation techniques [49]. It implements rigorous control of local truncation errors by automatic time-step selection. It delivers computational efficiency by automatically varying the integration order. 

The Selected Asymptotic Integration Method (5) has been utilized for many years at NRL for the solution of the coupled "stiff" ordinary differential equations associated with reactive flow problems. This program has been optimized for the ASC. 

The Matlab Simulink Model was designed to represent the model stmctuie and mass balance equations for SSF and is shown in Fig. 6. Shaded boxes represent the reaction rates, which have been lumped into subsystems. To solve the system of ordinary differential equations (ODEs) and to estimate unknown parameters in the reaction rate equations, the inter ce parameter estimation was used. This program allows the user to decide which parameters to estimate and which type of ODE solver and optimization technique to use. The user imports observed data as it relates to the input, output, or state data of the SimuUnk model. With the imported data as reference, the user can select options for the ODE solver (fixed step/variable step, stiff/non-stiff, tolerance, step size) as well options for the optimization technique (nonlinear least squares/simplex, maximum number of iterations, and tolerance). With the selected solver and optimization method, the unknown independent, dependent, and/or initial state parameters in the model are determined within set ranges. For this study, nonlinear least squares regression was used with Matlab ode45, which is a Rimge-Kutta [3, 4] formula for non-stiff systems. The steps of nonlinear least squares regression are as follows ... 

Petzold, L.R., 1983. Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations. SIAM J. Sci. Stat. Comput. 4, 136-148. 

A time scale separation method makes use of the fact that the physical and chemical time scales have only a limited range of overlap. The time scales of some of the more rapid chemical processes can thus be decoupled and be described in approximate ways by the Quasi Steady State Assumption (QSSA) or partial equilibrium approximations for the selected species. This reduces the species list to only the species left in the set of differential equations. Also, eliminating the fastest time scales in the system solves the numerical stiffness problem that these time scales introduce. Numerical stiffness arises when the iteration over the differential equations need very small steps as some of the terms lead to rapid variations of the solution, typically terms involving the fastest time scales. 

Other popular methods for numerical solutions of DEs are the Runge-Kutta methods. They again come in forms of different order, depending on the number of selected points on each sub-interval for which the function is evaluated and averaged. The development of these methods includes quite sophisticated analyses of errors (deviations from the true solutions) which occur with functions of different properties. A major problem in the numerical integration of rate equations is stiffness. A differential equation is called stiff if, for instance, different st s in the process occur on widely different time scales. It is very in dent to compute with time intervals suitable for the steepest part of the progress curve (see Press et al., 1986, chapter 16 and commercial programs recommended on p. 36). 

This selection process is then iterated, beginning from an initial state of the system, as defined by species populations, to simulate a chemical evolution. A statistical ensemble is generated by repeated simulation of the chemical evolution using different sequences of random numbers in the Monte Carlo selection process. Within limits imposed by computer time restrictions, ensemble population averages and relevant statistical information can be evaluated to any desired degree of accuracy. In particular, reliable values for the first several moments of the distribution can be obtained both inexpensively and efficiently via a computer algorithm which is incredibly easy to implement (21, 22), especially in comparison to now-standard techniques foF soTving the stiff ordinary differential equations (48, 49) which may arise in the deterministic description of chemical kinetics (53). Now consider briefly the essential features of a simple chemical model which illustrates well the attributes of stochastic chemical simulations. 

It is also possible to have differential-algebraic equations. There will be both differential and algebraic equations, with all the variables possibly occurring in all the equations. This essentially means that the variables determined from the algebraic equations are acting infinitely fast this is merely a system of stiff equations taken to an extreme. Special methods exist for these problems, too. Because some of the variables are supposed to change immediately, they cannot be included in the error tests used to select the time step. 

Because this model cannot be expressed as an integrated function, differential eqnations must be used. The selection of the appropriate advan (e.g., ADVAN 6, 8, or 9) should be examined carefnUy. It is not possible to determine a priori if a set of equations is stiff. Therefore, aU advans shonld be tested at least initially. Selection shonld be based on run times and minimization status. It should be noted that the selection of tol should be such that it is at least as large as nsig to improve chances of snccessful minimization. The compartments are defined here as well. 

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