Recall that, for the Taylor method of order 2, the function T
(2)(t, y) is defined as
T
(2)(t, y) = f(t, y) + h
2
∂f
∂t (t, y) + h
2
∂f
∂y (t, y)f(t, y), (1)
where we have used the chain rule to evaluate the derivative term. We want to derive a RungeKutta method by approximating T
(2)(t, y) in the form
a1f(t, y) + a2f(t + α1, y + β1f(t, y)), (2)
where a1, a2, α1, and β1 are parameters that we must choose.
(a) Taylor’s theorem in two variables implies that we may write
f(t + α1, y + β1f(t, y)) = f(t, y) + α1
∂f
∂t (t, y) + β1f(t, y)
∂f
∂y (t, y) + R1(t + α1, y + β1),
where R1(t + α1, y + β1) is the remainder term. If we want (2) to approximate (1), then we
must choose the parameters a1, a2, α1, and β1 such that the matching condition
f(t, y)+
h
2
∂f
∂t (t, y)+
h
2
f(t, y)
∂f
∂y (t, y) = a1f(t, y)+a2
f(t, y)+α1
∂f
∂t (t, y)+β1f(t, y)
∂f
∂y (t, y)
holds. By matching the coefficients on like-terms (color-coded for your convenience), derive
the three equations that involve the parameters a1, a2, α1, and β1 (but do not solve these
equations). Make sure to distribute a2 into the bracketed expression before matching!
(b) You will notice that there are four unknown parameters but only three equations, which
suggests that we have some flexibility in choosing the parameters. The standard approach
is to choose a2 freely and then solve for the remaining three parameters a1, α1, and β1. Do
so for the choice a2 = 1, and state the name of the Runge-Kutta method from lecture
that corresponds to these parameters.
(c) Now choose a2 =
2
3
and solve for the remaining parameters a1, α1, and β1. This choice of
parameters gives Ralston’s method. (Note: There is another method where you choose
a2 =
3
4
that is also called Ralston’s method, but we will go with the one above.
