RKF method for second order ODE

Hi Does anybody know where program in MATLAB for solution of 2nd order diff equation using RKF method is available. I've seen the RKF method for first order ODE in this site (from book by mathews & Fink). Bur is it available for second order ODE?

TIA

Answer 1

[Star Strider]

You have to create two first-order ODEs from your second order ODE. This is not difficult.

If you have the SymbolicMath Toolbox, it will even do it for you with the odeToVectorField function, then by matlabFunction to create a function from it that the ODE solvers can use.

[Star Strider]

Sreedhar, I looked online and can’t find any reference to the ‘Runge Kutta F method’. It may go by another name, but I can’t find any reference to it anywhere by that name. The Wikipedia article on Runge–Kutta methods doesn’t mention it.

From the ‘> Algorithms’ section under More About for ode45, both ode45 and ode23 are adaptive Runge-Kutta algorithms:

• ode45 is based on an explicit Runge-Kutta (4,5) formula, the Dormand-Prince pair. It is a one-step solver – in computing y(tn), it needs only the solution at the immediately preceding time point, y(tn-1). In general, ode45 is the best function to apply as a first try for most problems. [3]
• ode23 is an implementation of an explicit Runge-Kutta (2,3) pair of Bogacki and Shampine. It may be more efficient than ode45 at crude tolerances and in the presence of moderate stiffness. Like ode45, ode23 is a one-step solver. [2]

The MATLAB ode45 and ode23 functions are adaptive time-step solvers. I believe all the others are as well.

That’s the best I can do.

[Sreedhar]

Star strider Many thanks

[Star Strider]

My pleasure!

Answer 2

[Sreedhar]

Star Strider Thanks for the answer. I am able to turn the 2nd order ODE to 2 first order ODE's and solve it using ode45. But there is a problem during running which I feel can be resolved by RKF method as it uses adaptive time step. However, there appears to be no function built into MATLAB that uses RKF method. Any clues (textbook reference or MATLAB examples) on how to do this ?

TIA

Answer 3

[Meg Noah]

https://ntrs.nasa.gov/citations/19680027281

Classical Fifth-, Sixth-, Seventh-, and Eighth-Order Runge-Kutta Formulas with Stepsize ControlRunge-Kutta formulas of high order with stepsize control through leading truncation error term

Document ID

19680027281

Document Type

Other - NASA Technical Report (TR)

Authors

Fehlberg, E.(NASA Marshall Space Flight Center Huntsville, AL, United States)

Date Acquired

September 8, 2013

Publication Date

October 1, 1968

Subject Category

MATHEMATICS

Report/Patent Number

NASA-TR-R-287

Distribution Limits

Public

Copyright

Work of the US Gov. Public Use Permitted.

[Meg Noah]

Answer 4

[Meg Noah]

https://ntrs.nasa.gov/citations/19690021375

Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problemsLow order Runge-Kutta formulas with step control for heat transfer problems

Document ID

19690021375

Document Type

Other - NASA Technical Report (TR)

Authors

Fehlberg, E.(NASA Marshall Space Flight Center Huntsville, AL, United States)

Date Acquired

September 2, 2013

Publication Date

July 1, 1969

Subject Category

MATHEMATICS

Report/Patent Number

NASA-TR-R-315

Funding Number(s)

CONTRACT_GRANT: 129-04-03-00-62

Distribution Limits

Public

Copyright

Work of the US Gov. Public Use Permitted.