Adrian Down April 25, 2006
Second-order Runge-Kutta method
Last time, we began to develop methods to obtain approximate solutions to the diﬀerential equation x = f (t, x(t)) with error of O(h2 ), where h is the ˙ spacing of the mesh used to calculate the approximation. To construct these methods, our goal was to minimize the local truncation error. We saw that, in general, the global truncation error should be one order less in h. Last time, we introduced an approximation that generalized the Modiﬁed Euler method. The general form of the expression was, y(t + h) − y(t) = ω1 hF (t) + ω2 hf (t + αh, y(t) + βhF (t)) where F (t) ≡ f (t, y(t)). Our proposal was to choose ω1 , ω2 , α and β such that the local truncation error of the approximation is O(h3 ), from which we expect the desired global truncation error to be O(h2 ).
We began to evaluate this condition last time by Taylor expanding the function f up to second or..