The Euler Method

Numerical integration of sets of differential equations is a well developed field of numerical analysis, e.g. most engineering problems involve differential equations. Here, we only give a very brief introduction to numerical integration. We start with the Euler method, proceed to the much more useful Runge-Kutta algorithm and finally demonstrate the use of the routines that are part of the Matlab package. 

As usual, we start with a simple example consider the reversible reaction  

While there is an analytical solution for this mechanism, the formula for the calculation of the concentration profiles for A and B is fairly complex, involving the tan and atan functions (according to Matlab s symbolic toolbox). We use it to demonstrate the basic ideas of numerical integration. 

The main disadvantage of the Euler method is that the calculated approximation for the concentrations are systematically wrong for each step. 

In contrast to our preferred standard mode in this book, we do not develop a Matlab function for the task of numerical integration of the differential equations pertinent to chemical kinetics. While it would be fairly easy to develop basic functions that work reliably and efficiently with most mechanisms, it was decided not to include such functions since Matlab, in its basic edition, supplies a good suite of fully fledged ODE solvers. ODE solvers play a very important role in many applications outside chemistry and thus high level routines are readily available. An important aspect for fast computation is the automatic adjustment of the step-size, depending on the required accuracy. Also, it is important to differentiate between stiff and non-stiff problems. Proper discussion of the difference between the two is clearly outside the scope of this book, however, we indicate the stiffness of problems in a series of examples discussed later. So, instead of developing our own ODE solver in Matlab, we will learn how to use the routines supplied by Matlab. This will be done in a quite extensive series of examples. 

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The basic idea behind the Euler method is to set the change in w per increment of time as... 

The simplest method for integrating eq. (14.23) is the Euler method. A series of steps are taken in the direction opposite to the normalized gradient. 

The Euler method, while extremely inaccurate, is also extremely simple. This method is based on the definition of the derivative... 

Ordinary differential equations the Euler method Quite commonly, differential equations appear in the form... 

The Euler method has little practical value, but forms the basis for most of the more elaborate methods. It consists in a first-order expansion of the derivative. The approximation at step t[Pg.129]

Table 3.7. Solution of the differential equation dyjdt = — 2ty by the Euler method with a time...

Using Simpson s rule to evaluate the integral on the right-hand side of (24), SJG solved the equation by means of the Euler method. Although the technique is straightforward and efficient to apply for simple systems, it could prove more cumbersome for complicated three-dimensional systems and require the use of a more accurate method than the Euler one. 

The stability of the Euler method is improved by using interpolation instead of extrapolation, and considering the tangent evaluated at 4+1 ... 

The PES in the vicinity of IRC is approximated by an (N - -dimensional parabolic valley, whose parameters are determined by using the gradient method. Specific numerical schemes taking into account p previous steps to determine the (p + l)th step render the Euler method stable and allow one to optimize the integration step in Eq. (8.5) [Schmidt et al., 1985]. When the IRC is found, the changes of transverse normal vibration frequencies along this reaction path are represented as... 

It turns out that the order of this approximation is also the global error order of the calculation using the Euler method. An alternative way to proceed is to go from the Taylor expansion for y(t + St), as in (3.3),... 

What we have here, with the Euler method, is the definition of the derivative as pertaining to time t (or nfit) and thus f(y(t)) or /( . ) on the right-hand side. For our specific example (4.3), this becomes approximately... 

For the mathematics of this, consider the discrete equation resulting from the Euler method, as in (4.8). Note that the new point, yn+i is formed from the old point yn by the addition of a term, here St f(yn). With RK, these terms are given the symbols fcj, there are from one to several of them, and they are added in a weighted manner. The procedure is to generate a number of these k s. One begins with an Euler step,... 

Note that the Euler method can be regarded as a first-order RK form, if we write (4.8) as... 

For brevity, the Euler method will be treated as a special case of RK, considered as RK1. The method is then to start by calculating a vector of k values, one for each y element. Discretising directly from (4.51), this is... 

The numerical model-Simulator NV-Simulator V. At this point, we must find the more suitable variant for passing from the differential or partly differential model equations to the numerical state. For the case of the monodimensional model, we can select the simplest numerical method - the Euler method. In order to have a stable integration, an acceptable value of the integration time increment is recommended. In a general case, a differential equations system given by relations (3.55)-(3.56) accepts a simple numerical integration expressed by the recurrent relations (3.57) ... 

IRC path is of special importance in connection with studies of reaction dynamics, since Although it is clear that RK4 is more stable and accurate than the Euler method for a... 

The simplest method for integrating eq. (14.23) is the Euler method. A series of steps e taken in the direction opposite to the normalized gradient.------------------------------... 

Conversion for maximum mixedness. The Euler method will be used for numerical integration ... 

The spreadsheet in Figure 9-21 illustrates the use of the RK method to simulate the first-order kinetic process A - B with initial concentration [A]o = 0.2000 and rate constant k = 5 x 10 . The differential equation is d[A]t/dt = -k[A]t. This equation is one of the simple form dy/dx = F(y), and thus only the yi terms of Ti to T4 need to be evaluated. The RK terms (note that Ti is the Euler method term) are shown on the following page. 

Compare the RK result in column F of Figure 9-21 with the analytical expression for the concentration, [A] = [A]oe in column G. After one half-life (row 13) the RK calculation differs from the analytical expression by only 0.00006%. (Compare this with the 3.6% error in the Euler method calculation at the same point.) Even after 10 half-lives (not shown), the RK error is only 0.0006%. 

Although it is clear that RK4 is more stable and accurate than the Euler method for a given step size, this does not necessarily mean that it is the most efficient method. Since the RK4 method requires four gradient calculations for each step, the simple Euler can employ a step size four times as small for the same computational cost. Similarly,... 

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