Euler’s explicit method

While the examples given here have dealt only with chemical reaction kinetics, the method illustrates how one can, in general, solve single as well as coupled differential equations. Euler s explicit method is useful as a qualitative tool it is easily implemented, and can provide a reasonably close result when Af is sufficiently small. The latter requirement, however, may make the explicit method impractical on a spreadsheet. For quantitative work, an implicit method is usually required, as it provides a better approximation given the limited number of iterations practical on a spreadsheet. 

Let us pick up a value of At = 0.1 and carry out five successive iterations which will demonstrate how to implement Euler s explicit method. Finally, a comparison will be conducted between the approximate and analytical solutions. 

Obviously, the PRE diminishes when a value of At is sufficiently small. Hence, the smaller the At, the higher the accuracy of the approximate solution. On the other hand, the number of steps and function evaluations will increase as At is reduced, which means a longer computational (CPU) time. There is a trade-off between these two extremities. Table 7.1 compares Euler s explicit method with the analytical method at each step of integration, using the PRE as the indicator of goodness. 

For a given method, PRE depends on both At and t itself. Euler s explicit method can be superior to Euler s implicit method at low values of t and becomes inferior at high values of t. The same is true with At. 

The following code describes the solution for the first-order ODE using Euler s explicit method ... 

Using Euler s explicit method, calculate the PRE involved in the solution of ODE ... 

In section 9.2 we illustrated one explicit method, Euler s forward method. In the present section, we likewise used only one type of implicit method, based on the trapezoidal or midpoint rule. All our examples have used constant increments Af higher computational efficiency can oftenbe obtained by making the step size dependent on the magnitudes of the changes in the dependent variables. Still, these examples illustrate that, upon comparing equivalent implicit and explicit methods, the former usually allow larger step sizes for a given accuracy, or yield more accurate results for the same step size. On the other hand, implicit methods typically require considerably more initial effort to implement. 

Equations (1) and (2) are solved by Euler s explicit scheme using finite difference method. The acceleration of a particle can be obtained from the known contact forces, moments, mass and mass moment of inertia. The acceleration is integrated by time to deld the velocity increment and the velocity is integrated again by time to yield the displacement increment. By repeating these processes for all particles, the motion of all particles and the dynamical behavior of granular material can be obtained entirely. Consequently, the unbalanced forces and moments produce the linear and rotational accelerations of particle in the next calculating step successively. 

Table 7.1 The solution for dt using Euler s explicit and the anal5 ical method.

Runge-Kutta methods are explicit methods that use several function evaluations for each time step. Runge-Kutta methods are traditionally written for/(f, y). The first-order Runge-Kutta method is Euler s method. A second-order Runge-Kutta method is... 

Seader and Henley (1998) noted the following criteria for explicit Euler s method to prevent the instability and oscillations respectively. 

With an example of batch distillation, Seader and Henley showed that the time step needed in implicit Euler s method was 200 times of that needed for explicit Euler s method. 

Another way to study the stability of explicit equations is to use the positivity theorem. For Euler s method, the equations can be written in the form... 

Euler s and RK methods are also known as one-step techniques which use function values only in a single step, that is, in the preceding step. However, in the multistep techniques, evaluation of each step requires function values from more than one of the preceding steps. The benefit of the multistep techniques is the use of additional information to obtain more accurate solutions. The Adams-Bashforth methods for explicit solution of Equation 11.1 are multi-step in nature and are given in second and fourth orders in Equations 11.19 and 11.20, respectively, as follows ... 

Now there is no unstable behavior regarding the size of h (Figure 2.2). Note that in the explicit Euler s method we were approximating the solution by a polynomial, and there exists no pol3momial that can approximate the exponential term as x tends to 00, hence, the instability. By using the implicit method, we have expressed the solution in the form of a rational function, which can go to zero, as t tends to 00. 

For a step size of h = 10 5 s evaluate the left-hand side of this equation for species A, B, and C using the A s found in step 3 above. Is the explicit Euler method predicted to be stable for this stepsize hi... 

Depending on the numerical techniques available for solving optimal control or optimisation problems the model reformulation or development of simplified version of the original model was always the first step. In the Sixties and Seventies simplified models represented by a set of Ordinary Differential Equations (ODEs) were developed. The explicit Euler or Runge-Kutta methods (Huckaba and Danly, 1960 Domenech and Enjalbert, 1981) were used to integrate the model equations and the Pontryagin s Maximum Principle was used to obtain optimal operation policies (Coward, 1967 Robinson, 1969, 1970 etc.). 

The formula is known both as the improved Euler formula and as Heun s method [170]. This method has roughly the stability of the explicit Euler method. Unfortunately, the stability may not be improved by iterating the corrector because this iteration procedure converges to the trapezoid rule solution only if At is small enough. 

The Newton-Euler method is well suited to a recursive formulation of the kinematic and dynamic equations of motion (Pandy and Berme, 1988) however, its main disadvantage is that all of the intersegmental forces must be eliminated before the governing equations of motion can be formed. In an dtemative formulation of the dynamical equations of motion, Kane s method (Kane and Levinson, 1985), which is also referred to as Lagrange s form of D Alembert s principle, makes explicit use of the fact that constraint forces do not contribute directly to the governing equations of motion. It has been shown that Kane s formulation of the dynamical equations of motion is computationally more efficient than its counterpart, the Newton-Euler method (Kane and Levinson, 1983). 

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