1. Adaptive Runge-Kutta addresses the problem of functions that change rapidly at a point

2. Would like to use small size steps in the area of rapid change - normal size steps in area of normal change Two approaches behind adaptive step size look at difference between predictions with different step sizes but same order RK look at difference between predictions with different order RK

3. Step-halving or adpartive Runge-Kutta let y 1 be single-step prediction let y 2 be prediction using two half steps The correction is fifth order accurate

4. Example:

5. Integrate y’ from x=0 to 2 using h=2, and improve using adaptive RK Complete step results are Half step results are

6. The correction is The corrected value is Compare to true value y(2)= 2.524369

7. Runge-Kutta-Fehlberg Uses two different RK predictions of different order Special choice of methods lets you use results from 4th order in 5th order RK - then combine them

8. Fourth order RK Fifth order RK

9. Formula for k’s

10. Example Use h=2, and the RKF method

11. The results are RK4= 2.542811 RK5= 2.554121 and Ea=RK5-RK4=2.554121-2.542811=0.01131 Now adjust stepsize

12. If Ea is too small, increase step size If Ea is too large, decrease step size

13. Stiffness stiff equation involves rapidly changing parts and slowly changing parts Solution is

15. Look at homogeneous part of equation In general Explicit Euler’s method

16. Look at what happens to y over long time - stability Ifthen y goes to infinity Sofor explicit method to work - small h

17. Need to use implicit methods, rather than explicit Implicit form of the Euler method Can solve to get

18. Implicit Euleris always stable - as i increases y goes to 0

19. Example: Explict solution: since a is 2000, let h=0.0001

20. Stability limit is h=0.0005 Try h=0.0007

21. Try h=0.001

22. h=0.002

23. Implicit approach:

24. h=0.002