Solving ODEs using Runge Kutta Fehlberg Method (RK45)

This code solves a single ordinary differential equation (ODE) using Runge Kutta Fehlberg Method (RK45), that has an error estimator of order O(h⁵). Here you can use both RK4 and RK5, by default the code gives RK4 solution, to use RK5 solution uncomment the line 28.

Code Outputs:

• Chart comparison of the exact solution (if exist) with the numerical Solution.
• Error chart of Runge Kutta Fehlberg Method.
• Maximum Error Value of Runge Kutta Fehlberg Method.

Max. Error = 1.073e-10

Input Requirements:

• ODE you want to solve.
• Initial Condition.
• Exact solution (optional)

About the Method:

Runge–Kutta–Fehlberg method (or Fehlberg method) is an algorithm in numerical analysis for the numerical solution of ordinary differential equations. It was developed by the German mathematician Erwin Fehlberg and is based on the large class of Runge–Kutta methods.

The novelty of Fehlberg’s method is that it is an embedded method from the Runge–Kutta family, meaning that identical function evaluations are used in conjunction with each other to create methods of varying order and similar error constants. The method presented in Fehlberg’s 1969 paper has been dubbed the RKF45 method, and is a method of order O(h⁴) with an error estimator of order O(h⁵). By performing one extra calculation, the error in the solution can be estimated and controlled by using the higher-order embedded method that allows for an adaptive stepsize to be determined automatically.

Butcher Tableau for Fehlberg’s 4(5) Method:

Any Runge–Kutta method is uniquely identified by its Butcher tableau. The embedded pair proposed by Fehlberg:

The first row of coefficients at the bottom of the table gives the fifth-order accurate method, and the second row gives the fourth-order accurate method.

References:

[1] Erwin Fehlberg (1969). Low-order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems. NASA Technical Report 315.

[2] Erwin Fehlberg (1970). ‘Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme,’ Computing (Arch. Elektron. Rechnen), vol. 6, pp. 61–71. doi:10.1007/BF02241732

[3] Ernst Hairer, Syvert Nørsett, and Gerhard Wanner (1993). Solving Ordinary Differential Equations I: Nonstiff Problems, second edition, Springer-Verlag, Berlin. ISBN 3-540-56670-8.