Euler method accuracy

A high degree of accuracy is not called for in many calculations of the evolution of environmental properties because the mathematical description of the environment by a reasonably small number of equations involves an approximation quite independent of any approximation in the equations solution. Figure 2-3 shows how the accuracy of the reverse Euler method degrades as the time step is increased, but it also shows the stability of the method. Even a time step of 40 years, nearly five times larger than the residence time of 8.64 years, yields a solution that behaves like the true solution. In contrast, Figure 2-2 shows the instability of the direct Euler method a time step as small as 10 years introduces oscillations that are not a property of the true solution. 

The key feature of the systems to be considered in this book is that they have short memories that is, the effects of perturbations diminish with the passage of time. In the example of this chapter, the carbon dioxide pressure returns to a value of 1 within a century or two of the perturbation, regardless of the size of the initial perturbation. In this kind of system, computational errors do not grow as the calculation proceeds instead, the system forgets old errors. That is why the reverse Euler method is useful despite its simplicity and limited accuracy. The many properties of the environment that are reasonably stable and predictable can, in principle, be described by equations with just this kind of stability, and these are the properties that can be simulated using the computational methods described in this book. 

There are three major problems for the Euler method. First, the accuracy is poor, since the method is based upon Eq. (F.16), in which only a first-order difference expression is used. The errors in the method are proportional to At. Second, stability is difficult to achieve for many problems. The only way to have a stable Euler method is to use a small enough time step-size, but you may not know what value is sufficient. Furthermore, a value that is sufficient at the beginning may not be sufficient later on, and it may take an excessively long time to finish the computation. Third, to validate the results it is necessary to solve the problem at least twice, with different time-steps. The method can, however, be programmed in Excel, as Figures A.3 and A.4 demonstrate. 

Truncating the path expansion at second-order (after the third term) yields a simple quadratic approximation to the path. Determining the coefficient of the second order term requires second energy derivatives. So if force constants are available, then a quadratic step can be taken along the path. This quadratic step accounts for the curvature of the reaction path at the point of expansion and consequently allows a larger step size than the Euler method does for a given accuracy. 

For h < he (3.3) can be computed by the extrapolated explicit Euler-method of order k with relative accuracy e. 

This algorithm of numerical integration is called the Euler method. The error is proportional to At. A more detailed discussion of the Euler method, as well as methods with better accuracy (and concomitant higher complexity), can be found in any text on numerical methods. A fine text with many examples in the context of chemical engineering is Applied Numerical Methods by B. Carnahan, H. A. Luther, and J. O. Wilkes (Krieger Publ., Melbourne, FL, 1990). 

In order to give an introduction to the most important concepts used in different methods, such as accuracy, stability, and efficiency, we will start with the exphcit Euler method, which is the simplest numerical method to use when solving ODEs. At this point, we should stress that the accuracy of this method is low, and that it is only conditionally stable. For this reason, it is not used in practice to solve any engineering problems. However, it serves well to illustrate a method for solving ODEs numerically, and for introducing the concepts of accuracy and stability. Later in this chapter more advanced numerical methods will be presented methods that have higher accuracy and better stability properties, and that are implemented in modem software products. 

By using the globally first-order accurate Euler method, 0 h), the error is halved becomes when the step size is halved. If the step size is further reduced, the global error continues to decrease, but very slowly, as shown in Figure 6.3. A major drawback of the Euler method is that it has low accuracy a very small step size is required to control and keep error low. The errors, shown in Figure 6.3, have been calculated as the difference... 

Finally the solution converges at z = 0.916079783099616, which is the reactant concentration, y, predicted by the implicit Euler method at the first time step i = 0.1. This procedure is repeated for the following nine time steps to determine the final reactant concentration. To sum up, the implicit Euler method involves more computation it does not improve accuracy because it is only first-order accurate, but it significantly improves stability. 

The accuracy of the Euler method can be improved by utilizing a combination of forward and backward differences. Note that the first forward difference of y at i is equal to the first backward difference of y at (i + 1) ... 

This combination of steps is known as the Euler predictor-corrector (or modified Euler) method, whose application is demonstrated graplhcaliy in Fig. 5.4. Correction by Eq. (5.67) may be applied more than once until the corrected value converges, that is, the difference between the two con.secutive corrected values becomes less than the convergence criterion. However, not much more accuracy is achieved after the second application of the corrector. 

The explicit, as well as the implicit, forms of the Euler methods have error of order (h ). However, when used in combination, as predictor-corrector, their accuracy is enhanced, yielding an error of order (h ). This conclusion can be reached by adding Eqs. (5.57) and (5.63) ... 

We now use this formalism to demonstrate the derivation of a higher order integration method than the explicit Euler one. In the explicit Euler method we neglect the time-variation of a and b over the time step. This is particularly bad for the second integral as dWi is of order and thus the explicit Euler method is only 1/2-order accurate for predicting the actual trajectory. Thus, let us increase die order of accuracy of this term by using a t accurate expansion ofbmtk[Pg.345]

The last term is die leading-order correction to the explicit Euler method to raise the order of accuracy to 1. We next evaluate the stochastic integral... 

A combination of open- and closed-type formulas is referred to as the predictor-corrector method. First the open equation (the predictor) is used to estimate a value for y,, this value is then inserted into the right side of the corrector equation (the closed formula) and iterated to improve the accuracy of y. The predictor-corrector sets may be the low-order modified (open) and improved (closed) Euler equations, the Adams open and closed formulas, or the Milne method, which gives the following system... 

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