Second-Order RK Method

It is straightforward to derive the recurrence relations for the second-order RK method. Suppose that the trapezoidal rule is used to evaluate the integral on the right-hand side in Equation 5.20  

Since y +, is not known, consider approximating it by /(y +i,t +i) where y +, is a first estimate for calculated by the forward Euler method  

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Calculations for the second-order RK method are shown in Figure 5.5. Also shown are the analytical (exact) solution and that produced by the Euler method using the same h = 0.1 min. 

It is obvious that the second-order RK method produces significantly better results than those of the Euler method. Figure 5.5 also illustrates that it is straightforward to implement this method in Excel . Similar calculations using the fourth-order RK method are shown in Figure 5.6. 

Exercise 5.2 Solve the ODEs of Exercise 5.1 using the second-order RK method. Eind acceptable values of h when using the second-order RK method. 

Exercise 5.8 Implanent the second-order RK method for any number of simultaneous ODE-IVPs in VBA. Use Example 5.7 as a model for designing, coding, and testing the program. Solve the problem of Exercise 5.7 to test the program. 

The fourth-order RK method can be derived in a manner similar to that for the second-order method. The basic idea, once again, is to subdivide the interval h and to use successive approximations to y +i. The final is a weighted average of the individual approximations. The resulting equations are... 

Rate methods. As for reactions with a single reactant, the most primitive and convenient rate method is to guess and then adjust the reaction orders until an approximately straight-line plot of —rk versus the rate versus Ck CB. .. is obtained (see Table 3.1). For example, if a reaction is expected to be second order in A and first order in B, the rate would be plotted versus CACB. For first guesses of reaction orders, the available data can be subdivided so that the concentration of all but one participant are the same within each group. The dependence of the rate on the concentration of that participant then is an indication of the respective order. Often, the evaluation need not be carried beyond this stage. 

A more advanced method is the Runge-Kutte (RK). The idea here is to generate some intermediate steps which allow a better and more stable estimate of the next geometry -Tor a given step size. The second-order Runge-Kutte (RK2) method first calculates the... 

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 ... 

RK.m-The Runge-Kutta methods This function is capable of solving die set of differential equations by a second-, third-, fourth-, or fifth-order Runge-Kutta method. The formulas that appeared in Table 5.2 are used for calculating a Runge-Kutta solution of the differential equations. 

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