Runge-Kutta, fourth order, method

Numerical methods are required to integrate these coupled ordinary differential equations and to calculate the time-dependent molar density of each component in the exit stream of the first CSTR. Generic integral expressions are illustrated below. The Runge-Kutta-Gill fourth-order correct algorithm is useful to perform this task. 

Numerical methods such as the Runge-Kutta-Gill fourth-order correct integration algorithm are required to simulate the performance of a nonisothermal tubular reactor. In the following sections, the effects of key design parameters on temperature and conversion profiles illustrate important strategies to prevent thermal runaway. 

Fig. 2. The time evolution of the total energy of four water molecules (potential-energy details are given in [48]) as propagated by the symplectic Verlet method (solid) and the nonsymplectic fourth-order Runge-Kutta method (dashed pattern) for Newtonian dynamics at two timestep values.

A popular fourth-order Runge-Kutta method is the Runge-Kutta-Feldberg formulas (Ref. Ill), which have the property that the method is fourth-order but achieves fifth-order accuracy. The popular integration package RKF45 is based on this method. 

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... 

Pag), where y o mole fraction of A in bulk gas phase can be determined iteratively, yAi = mole fiaction of A in gas inlet. Equations (1) to (6) were solved using fourth order Runge-Kutta method [1, 8]. The value of enhancement factor, E, was predicted using equation of Van Krevelen and Hoftijzer [2]. 

Both POLYMATH and CONSTANTINIDES use this method and also a fourth order Runge-Kutta method. Other methods are available in other software, but these two are adequate for the present book. 

The situation is different for those readers who do not have access to Matlab and rely completely on Excel. In the following, we explain how a fourth order Runge-Kutta method can be incorporated into a spreadsheet and used to solve non-stiff ODE s. 

Fourth Order Runge-Kutta Method in Excel... 

The fourth order Runge-Kutta method is the workhorse for the numerical integration of ODEs. Elaborate routines with automatic step-size control are available in Matlab. 

Exercise 1. From the values of Table 1 and Eq.(lO), write a computer program using a fourth order Runge-Kutta or fifth order Runge-Kutta-Fehlberg method and reproduce Figures 2, 3, 4, 5. In order to check that the chaotic behavior has been reached, it is necessary to run the program with two initial conditions very close, for example ... 

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

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). 

The fourth-order Runge-Kutta method is applicable for a set of N first-order ODEs, where the functional form of the first derivative for each equation i is known ... 

Starting with an initial value of cA and given c,(t), Eq. (8-4) can be solved for cA(t + At). Once cA(t + At) is known, the solution process can be repeated to calculate cA(t + 2At), and so on. This approach is called the Euler integration method while it is simple, it is not necessarily the best approach to numerically integrating nonlinear differential equations. As discussed in Sec. 3, more sophisticated approaches are available that allow much larger step sizes to be taken but require additional calculations. One widely used approach is the fourth-order Runge Kutta method, which involves the following calculations ... 

The computer program PLUG51 employing the Runge-Kutta fourth order numerical method was used to determine the conversions and the compositions of the components. Applying the Runge-Kutta method, Equations 5-328 and 5-329 in differential forms are... 

Dimensionless parameter estimation The model s equations were solved using a fourth-order Runge-Kutta method. The dimensionless parameters estimation (Table I) was made as follows Only two of the dimensionless numbers Ai, Af, A2, A3, Ai and A5 are known directly. The parameter Ai can be estimated with reasonable accuracy since the catalyst surface area was measured independently ... 

First, without explaining the details [15], we will develop an Excel spreadsheet for the numerical integration of the reaction mechanism 2as seen in Figure 7.13. The fourth-order Runge-Kutta method requires four evaluations of concentrations and derivatives per step. This appears to be a serious disadvantage, but as it turns out, significantly larger step sizes can be taken for the same accuracy, and the overall computation times are much shorter. We will comment on the choice of appropriate step sizes after this description. 

The method is one way to handle a stiff set of odes, and is an extension of fourth-order explicit Runge-Kutta. The function to be solved is approximated over the next time interval by a combination of a linear function of the dependent variable and a quadratic function of time (assuming that it is strongly time-dependent) and this increases the accuracy and stability of the fourth-order Runge-Kutta method considerably. Today, however, we have other methods of dealing with stiff sets of odes, so this method might be said to have outlived its usefulness. 

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