Runge-Kutta-Fehlberg

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 ... 

From Eq.(55) by using the Runge-Kutta or Runge-Kutta-Fehlberg intre-gration method the state variables x(t) can be calculated. 

Various integration methods were tested on the dynamic model equations. They included an implicit iterative multistep method, an implicit Euler/modified Euler method, an implicit midpoint averaging method, and a modified divided difference form of the variable-order/variable-step Adams PECE formulas with local extrapolation. However, the best integrator for our system of equations turned out to be the variable-step fifth-order Runge-Kutta-Fehlberg method. This explicit method was used for all of the calculations presented here. 

The method of characteristics, the distance method of lines (continuous-time discrete-space), and the time method of lines (continuous-space discrete-time) were used to solve the solids stream partial differential equations. Numerical stiffness was not considered a problem for the method of characteristics and time method of lines calculations. For the distance method of lines, a possible numerical stiffness problem was solved by using a simple sifting procedure. A variable-step fifth-order Runge-Kutta-Fehlberg method was used to integrate the differential equations for both the solids and the gas streams. 

The Runge-Kutta-Fehlberg is a further modification of the Runge-Kutta fourth-order method. It uses a fifth function evaluation to determine the appropriate step size. This method appears to be very efficient for non-stiff systems of differential equations. Additional details regarding this method and a computer listing can be found in a report by Fehlberg and in the chapter by Watt and Shampine. ... 

Runge-Kutta Fehlberg method (5th order) from 14. Fehlberg I, Fehlberg 4th, Kutaa-Nystrom, England II and England I have also been tested, but... 

Runge-Kutta Fehlberg I method (5th order) from 14. 

The Runge-Kutta Fehlberg and Runge-Kutta Fehlberg I have the same behavior and are worse than the new developed methods but better than all the other compared methods. 

In 33 the authors have developed a family of trigonometrically fitted Runge-Kutta methods for the numerical integration of orbital problems. The developed method is based on the Runge-Kutta Fehlberg I method (see 14). More specifically the new method is developed in order to integrate exactly any linear combination of the functions ... 

One of the most popular embedded Runge-Kutta methods is the fourth-order Runge-Kutta-Fehlberg method controlled by a fifth-order one. The equations that define it are as follows ... 

The Runge-Kutta-Fehlberg variant is usually slightly better than the other variants. 

BzzOdeRKF. Fourth/fifth-order Runge-Kutta-Fehlberg method. 

In [208] the authors obtained a numerical scheme and code for estimating the deposition of energy and momentum due to the neutrino pair annihilation (v -f- V e + c+) in the vicinity of an accretion tori around a Kerr black hole. In order to solve the collisional Boltzmann equation in curved space-time, the authors solved approximately the so-called rendering equation along the null geodesics. They used the Runge-Kutta Fehlberg... 

As is indicated in the main program, other integrators (such as the Euler or Runge-Kutta-Fehlberg) could have been used to solve this problem. That approach would provide a means to compare the results from each method. However, in the present example the classical Runge Kutta method was used and the results are compared with those from the analytical solution under the column labeled u(exact). The other columns, u(num) and diff, are the numerical results and the difference between the exact and the numerical. 

Runge-Kutta-Fehlberg (RKF) is another method involving fourth- and fifth-order integration formulations (Equations 11.11 and 11.12, respectively) whose difference can be... 

T. E. Simos, A Runge-Kutta Fehlberg method with phase-lag of order infinity for initial value problems with oscillating solution. Computers and Mathematics with Applications, 1993, 25, 95-101. 

Another method that uses fourth- and fifth-order embedded pairs is the Dormand-Prince method. The Dormand-Prince method is more accurate than the Runge-Kutta Fehlberg method and it is used by the MATLAB ode45 solver. Both methods have in common that the difference between the fourth- and fifth-order accurate solutions is calculated to determine the error, and to adapt the step size. The error estimate, e +, for the step is... 

Another method of controlling the step size is to obtain an estimation of the truncation error at each interval. A good example of such an approach is the Runge-Kutta-Fehlberg method (.see Table 5.2), which provides the estimation of the local truncation error. This error estimate can be easily introduced into the computer program, and let the program automatically change the step size at each point until the desired accuracy is achieved. 

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