Runge-Kutta Methods: Modified Euler II

Midpoint Method is ?1 = f(ti,X(ti)) ?2 = f(ti+0.5*h,X(ti+0.5*h)) Xi+1 = Xi + h * ?2

ti+1

Xi = Yi

Exact Yi+1

?2 Approximate Tangent at Midpoint

Approximate Tangent at Midpoint through ti

?1 Tangentat ti

Midpoint Approximation Xi+1

Global Error is O(h2))Second Order Method

Midpoint Method is ?1 = f(ti,X(ti)) ?2 = f(ti+0.5*h,X(ti+0.5*h)) Xi+1 = Xi + h * ?2