Numerical Methods To Solve Initial Value Problems

1. Numerical Methods To Solve Initial Value Problems An Over View of Runge-Kutta Fehlberg and Dormand and Prince Methods. William Mize

2. Quick Refresher • We are looking at Ordinary Differential Equations • More specifically Initial Value Problems • Simple Examples: • Solution of: • Solution of:

3. A Problem • How practical are analytical methods? • Equation: • We chose to find a Numerical solution because • Closed-form is to difficult to evaluate • No close-form solution

4. Some Quick Ground work • First Start with Taylor Series Approximations • Then Move onto Runge-Kutta Methods for Approximations • Lastly onto Runge-Kutta Fehlberg and Dormand and Prince Methods for Approximation and keeping control of error

5. How these Methods Work • All of the Methods will be using a step size method. • Error is determined by the size of step, order, and method used. • When actually calculating these, almost always done via computer.

6. Taylor Series Methods(Brief) • Taylor Series As Follows • Most Basic is Euler’s Method • Higher Order Approximations better Accuracy • But at a cost • What can we do?

7. Runge-Kutta Methods • Named After Carl Runge and Wilhelm Kutta • What they do? • Do the same Job as Taylor Series Method, but without the analytic differentiation. • Just like Taylor Series with higher and higher order methods. • Runge-Kutta Method of Order 4 Well accepted classically used algorithm.

8. Runge-Kutta of Order 2 • We don’t want to take derivatives for approximations • Instead use Taylor series to create Runge-Kutta methods to approximate solution with just function evaluations. • We Want to Approximate this with • Find A, B, C • We get: • Error

9. Runge-Kutta of Order 4 Error of Order

10. So What's next? • Already Viable Numerical Solution established what's the next step? • We want to control our Error and Step size at each step. • These methods are called adaptive. • Why? • Cost Less • Keep within Tolerance • Also look for More efficient ways of doing these things. • 10 Function Evaluation for RK4 and RK5 • Just 6 for RKF4(5)

11. Runge-Kutta Fehlberg • Coefficients are found via Taylor expansions

12. Next Step to find These Coefficients

13. Further Deriving • We assume =1

14. More and more… • So this was way more complicated than I actually thought it would be. • But it’s all leading some where! • Eventually we want to have all the in terms of • From there was must figure out our and

15. How to find • First Take coefficients from the 5th order equation. • Which ultimately leads to • Where we chose = 1/3 and = 3/8

16. = 1/3

17. = 3/8

18. Comparison(Problem)

19. Comparisons of Methods

20. Dormand and Prince Methods

21. Visual Comparison of Methods

22. Conclusion • Taylor’s method uses derivatives to solve ODE • RK uses only a combination of specific function evaluations instead of derivatives to approximate solution of the ODE • RKF is beneficial because you can control your step size so you have your global error within a predetermined tolerance • RK4 and RK5 uses 10 function evaluations vs RKF just 6 • Runge-Kutta Fehlberg is widely accepted and used commercially(Matlab, Mathematica, maple, etc)

23. Sources • Numerical Mathematics and Computing. Sixth Edition; Ward Cheny, David Kincaid • Low-Order classical Runge-Kutta Formulas with StepSize Control and their Application to some heat transfer problems. By Erwin Fehlberg(1969) • A family of embedded Runge-Kutta Formulae. By Dormand and Prince(1980)