Euler integration method

Starting with an initial value of and knowing c t), Eq. (8-4) can be solved for c t + At). Once c t + At) is known, the solution process can be repeated to calciilate c t + 2At), and so on. This approach is called the Euler integration method while it is simple, it is not necessarily the best approach to numerically integrating nonlinear differential equations. To achieve accurate solutions with an Eiiler approach, one often needs to take small steps in time. At. A number of more sophisticated approaches are available that allow much larger step sizes to be taken but require additional calculations. One widely used approach is the fourth-order Bunge Kutta method, which involves the following calculations ... 

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The implicit Euler integration method is examined here. We will use the set of two sequential reactions, rate constants, and initial conditions described in the previous problem. Note This problem uses results from tasks 1-3 in the previous problem.)... 

The same equation (1.79) approximated with the Euler integration method, gives... 

The Euler integration method transforms equation (4.15) into a difference equation with the form... 

The explicit Euler integration method is simply the linear extrapolation from the point at f to r +, using the slope of the curve at t . Figure 7.7 shows this Euler method with the local and global errors. 

Fig. 5.6 illustrates the propagation of error in the Euler integration method. Starting with a known initial condition yo, the method calculates the value y, which contains the truncation error for this step and a small roundoff error introduced by the computer. The error has been magnified in order to illustrate it more clearly. The next step starts with y, as the initial point and calculates yj. But because y, already contains truncation and roundoff errors, the value obtained foryj contains these errors propagated, in addition to the new truncation and roundoff errors from the second step. The same process occurs in subsequent steps. 

The two inequalities (5.184) and (5.186) describe the circle with a radius of unity on the complex plane shown in Fig. 5.7. Since the explicit Euler method can be categorized as a first-order Runge-Kutta method, the corresponding curve in this figure is marked by R K1. The set of values of hk inside the circle yields stable numerical solutions of Evq. (5.170) using the Euler integration method. 

Implicit Methods By using different interpolation formulas involving y, it is possible to cferive imphcit integration methods. Implicit methods result in a nonhnear equation to be solved for y so that iterative methods must be used. The backward Euler method is a first-order method. 

This set of ordinaiy differential equations can be solved using any of the standard methods, and the stability of the integration of these equations is governed by the largest eigenvalue of AA. If Euler s method is used for integration, the time step is hmited by... 

V. The auxiliary equation is normally an algebraic equation rather than an ODE. In chemical engineering problems, it will usually be an equation of state, such as the ideal gas law. In any case, the set of ODEs can be integrated numerically starting with known initial conditions, and V can be calculated and updated as necessary. Using Euler s method, V is determined at each time step... 

The integration of the state equations (Equation 10.21) by the fully implicit Euler s method is based on the iterative determination of x(t1+i). Thus, having x(t,) we solve the following difference equation for x(t, i). 

Figure 10.1 Schematic diagram of the sequential solution of model and sensitivity equations. The order is shown for a three parameter problem. Steps l, 5 and 9 involve iterative solution that requires a matrix inversion at each iteration of the fully implicit Euler s method. All other steps (i.e., the integration of the sensitivity equations) involve only one matrix multiplication each.

The concentration of each chemical species, as a function of time, during cure can be calculated numerically from Equations 3-6 using the Euler-Romberg Integration method if the initial concentrations of blocked isocyanate and hydroxyl functionality are known. It is a self-starting technique and is generally well behaved under a wide variety of conditions. Details of this numerical procedure are given by McCalla (12). 

Each trial curve generated by the Euler-Romberg integration method is similarly normalized by dividing each of its points by a value corresponding to the time at which the experimental curve reached its maximum. 

If Euler s method is used for integration, the time step is limited by... 

Much better results are obtained by using the parallel surface method (Section III.F.3), because the integral methods are used to determine (K). Nevertheless, PSM gives only approximate estimation of the Euler characteristic and is extremely time-consuming in comparison to the methods described below. 

The 1/r solution is in fact just an Euler s method approximation to the integral for the PFTR, in which one approximates the integral as a summation. The calculation is not very accurate because we used a 0.2 moles/liter step size to keep the spreadsheet small, but it illustrates the method and the identity between Euler s method and a spreadsheet solution. 

There are more refined one-step integrators than Euler s method. Multi-step integration methods use more than just one previously computed y and F value and thereby they can better account for y s curvature and higher derivatives. Thus they follow the actual solution curve much closer. But in turn, they need more supporting F data computations. 

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