Runge-Kutta Methods

• We can do better by symmetrizing derivative:

  • Take a trial step to midpoint, evaluate yn+1/2 and tn+1/2.
  • Use these to evaluate derivative f'(tn+1/2,yn+1/2).
  • Then use this to go back and take a full step.

• Thus:

  • yn+1 = yn + h f'(tn + ½ h,yn + ½ h f'(tn,yn)) + O(h3)

• Can show that O(h2) terms "cancel," so leading error term is O(h3)  2nd-order Runge-Kutta.

Notes: