Differential equations predictor-corrector method

M71 Solution of ordinary differential equations predictor-corrector method of Milne 7100 7188... 

Rigorous and stiff batch distillation models considering mass and energy balances, column holdup and physical properties result in a coupled system of DAEs. Solution of such model equations without any reformulation was developed by Gear (1971) and Hindmarsh (1980) based on Backward Differentiation Formula (BDF). BDF methods are basically predictor-corrector methods. At each step a prediction is made of the differential variable at the next point in time. A correction procedure corrects the prediction. If the difference between the predicted and corrected states is less than the required local error, the step is accepted. Otherwise the step length is reduced and another attempt is made. The step length may also be increased if possible and the order of prediction is changed when this seems useful. 

Predictor-corrector methods [47,48] are appropriate for the integration because they require only one evaluation of the slope for each integration step. Molecular simulation is unusual in the context of the numerical treatment of differential equations, because an approximation to the slope is available before the simulation is complete. This information can be used to update the state point as the simulation proceeds. An increment in a typical GDI series entails the following steps, which for concreteness we describe for an integration in the P-p plane... 

A nh). This extra effort may well be rewarded by the increased accuracy of the solution obtained. A more effective approach, which is frequently used in modern software packages, is to combine an explicit method with an implicit one to give a predictor-corrector technique. An explicit formula like eq. (7.5) or eq. (7.6) is used, not to give a final result, but to provide an initial guess of the solution, Ap nh), at f = nh. We can then insert this value of A into the differential equation (7.4) to obtain the predicted value of the derivative A p nh)= f Ap nh)). These values of A and A are then used in the implicit method to yield the final, corrected solution A nh). A secondary advantage of predictor-corrector methods is that the difference between Ap and A provides a reliable estimate of the error, typically (Acton, 1970),... 

In the methods explained so far, to solve a differential equation over an interval (xj, x,+i) only the value of y at the beginning of the interval was required. In the predictor corrector methods, however, four prior values are required for finding the value of y at x,+. A predictor formula is used to predict the value of y at x,+i and then a corrector formula is applied to improve this value. We now explain one such method. 

The total number of sensitivity equations is equal to the number of state variables times the number of model parameters (i.e. 10x22). These equations are derived in reference 13. The variation of the sensitivity coefficients along the reactor is determined from the numerical integration of the sensitivity differential equations (24) and model equations (1)-(10). It should be noted that the sensitivity equations (24) are extremely stiff. Thus, extra care must be taken in integrating these equations. Accordingly, a multi-step predictor corrector method suitable for stiff differential equations was used. 

The differential equations which arise in almost all chemical kinetic studies of complex reaction schemes are "stiff differential equations". In chemical kinetics the stiffness is caused by the huge differences in the reaction rate constants of the various elementary reactions. It is impossible to solve such a system of differential equations by the usual Rung-Kutta methods. Therefore we used a program, described by Gear (J ) as a special multi step predictor-corrector method with self adjusting optimum step size control. 

MEULER Solves a set of ordinary differential equations by % the modified Euler (predictor-corrector) method. 

X,Y]=MEULER( F, XI,XP,H,YI) solves a Set of ordinary differential equations by the modified Euler (the Euler predictor-corrector) method, from XI to XF. 

Solution of a set of nonlinear ordinary differential equations by the Adams-Moulton predictor-corrector method. 

The numerical solution of the primary circuit is carried out in two stages. In the first stage, IHX flow, pump flows, pump speeds and the sodium levels are calculated utilizing a standard ordinary differential equation solver based on the Hamming s predictor-corrector method. 

The solution of Equations 47, 48, and 49 requires numerical techniques. For such nonlinear equations, it is usually wise to employ a simple numerical integration scheme which is easily understood and pay the price of increased computational time for execution rather than using a complex, efficient, numerical integration scheme where unstable behavior is a distinct possibility. A variety of simple methods are available for integrating a set of ordinary first order differential equations. In particular, the method of Huen, described in Ref. 65, is effective and stable. It is self-starting and consists of a predictor and a corrector step. Let y = f(t,y) be the vector differential equation and let h be the step size. 

To integrate the ordinary differential equations resulting fi om space discretization we tried the modified Euler method (which is equivalent to a second-order Runge-Kutta scheme), the third and fourth order Runge-Kutta as well as the Adams-Moulton and Milne predictor-corrector schemes [7, 8]. The Milne method was eliminated from the start, since it was impossible to obtain stability (i.e., convergence to the desired solution) for the step values that were tried. 

Example 5.3 Solution of Nonisothermal Plug-Flow Reactor. Write general MATLAB functions for integrating simultaneous nonlinear differential equations using the Euler, Euler predictor-corrector (modified Euler), Runge-Kutta, Adams, and Adams-Moulton methods. Apply these functions for the solution of differential equations that simulate a nonisotherm plug flow reactor, as described below. ... 

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