Fourth-Order RK Method

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 

This method can be shown to be OQf) and is perhaps the most popular method for solving ODE-IVPs numerically. 

Suppose chemical A is in solution in a perfectly stirred tank and its concentration is C°(g/L). The constant volumetric flow into and out of the tank is F (L/min) and the tank volume is V(L). A mass balance on component A leads to 

FIGURE 5.5 Second-order RK results for mixing tank. 

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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. 

From Figure 5.6, it can be seen that the fourth-order RK method produces essentially exact results for this problem. As previously stated, the overall error associated with the fourth-order RK method is >(/) ). The method is easy to implement in Excel since it is an explicit method. 

Exercise 5.11 Implement the fourth-order RK method for any number of simultaneous ODE-IVPs in VBA. 

Recall the algorithm for using the fourth-order RK method with only one ODE ... 

This algorithm requires three evaluations of the derivative function at different times corresponding to the initial time, the mid point and the final time points wifli approximations to the solution values used at the three points. In all of fliese algorithms the derivative terms are evaluated in the order presented in the equation set The most popular RK methods are fourth order. Again there are an infinite number of possible forms, but the most common is the classical fourth order RK method given by this set of equations Wi =fiynh)... 

A more accurate technique used for numerical integration is the classical fourth-order Runge-Kutta (RK) method. In this... 

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 ... 

We will begin the learning of the integrators with the rkfixed function. It implements the fourth-order Runge-Kutta method (rk) with fixed step of integraticai (fixed). According to Mathcad syntax this function has five required arguments ... 

AdamsMoulton.m The Adams-Moulton method This function solves the set of differential equations using Eqs. (5.96) and (5.97). The required starting points are evaluated by the fourth-order Runge-Kutta method (using the function RK.m), which has the same order of truncation error as the Adaras-Moulton method. 

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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