The Runge-Kutta Method

The procedure for deriving the Runge-Kuttamethods can be divided into five steps which are demonstrated below in the derivation of the second-order Runge-Kutta formulas. 

Step 1 Choose the value of m, which fixes the accuracy of the formula to be obtained. For second-order Runge-Kutta, m = 2. Truncate the series (5.79) after the (m + 1) term  

Step 2 Replace each derivative of y in (5.80) by its equivalent in/, remembering that/is a function of both x and y(x)  

Step 5 In order for Eqs. (5.83) and (5.88) to be identical, the coefficients of the corresponding terms must be equal to one another. This results in a set of simultaneous nonlinear algebraic equations in the unknown constants Wj, Cj, and Oj,. For this second-order Runge-Kutta method, there are three equations and four unknowns  

For the second-order Runge-Kutta method, which we are currently deriving, let us choose Cj = 1 The rest of the parameters are evaluated from Eqs. (5.89)  

The Rimge-Kutta methods for numerical solution of the differential equation dy/dx = F(x, y) involve, in effect, the evaluation of the differential function at intermediate points between xi and Xj+i. The value of yi+ is obtained by appropriate summation of the intermediate terms in a single equation. The most widely used Runge-Kutta formula involves terms evaluated at X(, Xj + Ax/2 and X + Ax. The fourth-order Runge-Kutta equations for dy/dx = F(x, y) are 

If more than one variable appears in the expression, then each is corrected by using its own set of Ti to T4 terms. 

In essence, the fourth order Runge-Kutta method performs four calculation steps for every time interval. In the solution by Euler s method, decreasing the time increment to 5 seconds, to perform four times as many calculation steps, still only reduces the error to 0,9% after 1 half-life. 

Write down the differential equations that describe the system. 

From the set of differential equations, enter formulas in spreadsheet cells using Euler s method, Fill Down and make sure that the results make sense. 

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This improved procedure is an example of the Runge-Kutta method of numerical integration. Because the derivative was evaluated at two points in the interval, this is called a second-order Runge-Kutta process. We chose to evaluate the mean derivative at points Pq and Pi, but because there is an infinite number of points in the interval, an infinite number of choices for the two points could have been made. In calculating the average for such choices appropriate weights must be assigned. 

More than two points can be used in the Runge-Kutta method, and the fourth-order Runge-Kutta integration is commonly employed. Obviously computers are... 

The Runge-Kutta method takes the weighted average of the slope at the left end point of the interval and at some intermediate point. This method can be extended to a fourth-order procedure with error 0 (Ax) and is given by... 

The simulated concentration-time values are displayed graphically in Fig. 5-1. These values could not have been obtained from analytical expressions. Most schemes that comprised a few steps can be handled by the Runge-Kutta methods, provided the time scales of the various steps are comparable. 

There are four equations in four dependent variables, a, d, T, and T. They can be integrated using the Runge-Kutta method as outlined in Appendix 2. Note that they are integrated in the reverse direction e.g., a = at) — similarly for 2 and in Equations (2.47). 

The radial equations was then solved using the Runge-Kutta method (7). 

More complicated numerical methods, such as the Runge-Kutta method, yield more accurate solutions, and for precisely formulated problems requiring accurate solutions these methods are helpful. Examples of such problems are the evolution of planetary orbits or the propagation of seismic waves. But the more accurate numerical methods are much harder to understand and to implement than is the reverse Euler method. In the following chapters, therefore, I shall show the wide range of interesting environmental simulations that are possible with simple numerical methods. 

The higher order ODEs are reduced to systems of first-order equations and solved by the Runge-Kutta method. The missing condition at the initial point is estimated until the condition at the other end is satisfied. After two trials, linear interpolation is applied after three or more, Lagrange interpolation is applied. 

Figure 3.6 Numerical solution of ordinary differential equations sketch of the four steps of the Runge-Kutta method to the order four giving the n+1 th estimate y(n+1) from the nth estimate y ". ...

Table 3.8. Solution of the differential equation dy/dt = —2ty by the Runge-Kutta method to the...

The Runge-Kutta method for a system of differential equations... 

The single-variable method can be extended to a system comprising any number of differential equations. The Runge-Kutta method is commonly found in software packages. For simplicity, it will be described for a system of only two equations... 

These final equations can be compared with the equivalent in the Euler approach, equation (3.81). In the Runge-Kutta method a weighted average of 4 different approximations for the derivatives is used instead of the derivative at the beginning of the time interval. 

Equation (101) is numercially solved by the Runge-Kutta method. The agreement between the experimental values and theory is seen from Fig. 17, where the solid lines indicate the values obtained by theory. The liquid used for the data presented in Fig. 17 had n = 0.617 and k = 0.0059. 

The volume VF is the value of T for x = r/6. This is obtained numerically by using the Runge-Kutta method. The results are shown as dotted lines in Fig. 26, where they show excellent agreement in the range investigated. For calculation purposes, the value of X has been used as the mean of the upper and the lower bounds found by Wasserman and Slattery (W2). 

Equations (7) to (14) are solved numerically according to the Runge-Kutta method [145], with initial conditions as defined in equation (16). 

Analytical solution of the equations are not available, but these can be solved numerically by using IMSL subroutine IVPRK, which is a modified version of subroutine DVERK based on the Runge-Kutta method. The solution would provide exit concentration of the solute from which percent extraction can be obtained. 

Equation (2.53) can be solved numerically by the Runge-Kutta method (see Appendix 2A). Explicit solutions of (2.53) are given here. 

The most commonly used the Runge-Kutta method is that of fourth order, consisting of the following algorithm ... 

The numerical result from the Runge-Kutta method (0.979795) using step Ax = 0.2 is different from the exact solution (0.979796) only in the sixth digit. 

Equation (3.71) can be solved numerically using the Runge-Kutta method. Analytical solutions are also presented here. When all b —0, Eq. (3.71) collapses to Eq. (3.55). When A(, 0, there are three cases and the solutions to Eq. (3.71)depend on the discriminant of the cubic polynomial in the denominator of the right-hand side of Eq. (3.71),... 

To control the step size adaptively we need an estimate of the local truncation error. With the Runge - Kutta methods a good idea is to take each step twice, using formulas of different order, and judge the error from the deviation between the two predictions. Selecting the coefficients in (5.20) to give the same a j and d values in the two formulas at least for some of the internal function evaluations reduces the overhead in calculation. For example, 6 function evaluations are required with an appropriate pair of fourth-order and fifth-order formulas (ref. 5). 

Table III presents the values of the constants used in the calculations. The t/o data have been obtained from the variation of Pq with Z by numerical integration of the Gibbs-Duhem equation using the Runge-Kutta method (22,23). The comparison of the Pq values of Table I with those obtained by some previous workers (24, 25) shows that our results are at most higher by 0.5-1 Torr.

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