Euler integration

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 ... 

Example 2.15 Develop an extrapolation technique suitable for the first-order convergence of Euler integration. Test it for the set of ODEs in Example 2.3. 

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Euler integration is extremely simple to program, as will be illustrated in Example 4,3. This simplicity is retained, even as the number of ODEs increases and as the derivative functions become more complex and nonlinear. 

These step sizes scale directly with the time constant t. If t were 10, we could take steps that were 10 times bigger. So the maximum stable step size for the Euler integration is twice the time constant. 

Effect of integration step size with Euler integration. 

Therefore my experience has been that, for most of the complex systems that chemical engineers have to deal with, a simple Euler integration is just as good as, if not better than, the more complex fourth-order Runge-Kutta. 

Then to step to the next point in time, using Euler integration with a step size DELTA,... 

Table 5.5 gives values of parameters and steady state conditions. The variables with overscores or bars over them are steadystate values. Note that the time basis used in this problem is hours. Table 5.6 gives a FORTRAN program that simulates this system using Euler integration. The right-hand sides of the... 

Use of the explicit Euler integration scheme will be illustrated by integrating the following set of sequential reactions... 

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.)... 

Nonlinear Dynamic Simulation The nonlinear ordinary differential equations are numerically integrated in the Matlab program given in Figure 4.2. A simple Euler integration algorithm is used with a step size of 2 s. The effects of several equipment and operating parameters are explored below. 

The explicit Euler integration technique involves specifying the integration step size, At ... 

Two-dimensional Model. The same strategy has been used for a two-dimensional model. The mass and energy balances Eqs. 6-8 and Eqs. 10-13,have been integrated by a Crank-Nicolson procedure. After completing the calculation a new distribution of activity is evaluated from Eq. 9 by an explicit Euler integration. 

Schlick and Olson recently developed such an algorithm that permits larger time steps. The implicit Euler integration scheme is combined with the Langevin dynamics formulation, which contains frictional and random... 

Verification of the model. Several assumptions were made in section 3.1 which led to Eqs.(3-6), (3-8) to (3-10) for the determination of p>jj and pjk. For the reactions given by Eqs.(3-13) the results are summarized in the matrix given by Eq.(3-17). The validity of the results will be tested by writing the Euler integration algorithm for the differential equations, Eqs.(3-12), which describe the reaction mechanisms. 

The above equations are based on Eqs.(2-23) and (2-24). The justification for applying the above equations to flow systems under consideration lies in the complete agreement obtained by the Euler integration of the linear equations, Eq.(4-... 

It should be emphasized that the matrix representation becomes possible due to the Euler integration of the differential equations, yielding appropriate difference equations. Thus, flow systems incorporating heat and mass transfer processes as well as chennical reactions can easily be treated by Markov chains where the matrix P becomes "automatic" to construct, once gaining enough experience. In addition, flow systems are presented in unified description via state vector and a one-step transition probability matrix. 

The most computationally intensive step in statistical or dynamical studies based on reaction path potentials is the determination of the MEP by numerical integration of Eq. (2) and the evaluation of potential energy derivatives along the path, so considerable attention should be directed toward doing this most efficiently. Kraka and Dunning [1] have presented a lucid description of many of the available methods for determining the MEP. Simple Euler integration of Eq. 

If the model of the system to be simulated can be reduced to the form of equations (2.22), then a timemarching, numerical solution becomes possible by repeated application of, for instance, the first-order Euler integration formula ... 

Assuming Euler integration, and using a subscript terminology, the mass in the tank at the jth time instant may be calculated from the mass and the flows at the (j — l)th time instant ... 

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