MATH/APPM 4650
Class notes, March 18

Runge-Kutta methods

Today's notes will mostly be on the board.

So far:

• Euler's method, with local truncation error O(h): w₀ = y₀,    w_i+1 = w_i + h f(t_i, w_i)
• Second-order Taylor method, with local truncation error O(h): w₀ = y₀,    w_i+1 = w_i + h f(t_i, w_i) + h²/2 [f_t(t_i, w_i) + f(t_i, w_i) f_y(t_i, w_i)]
• n^th-order Taylor method involves computing derivatives up to y⁽ⁿ⁾(t) in terms of t and y(t). Has local truncation error O(h).

Problem: f may not be known very accurately, so derivatives will be known to much less accuracy.
Even if f were known exactly, the derivatives might be hard to compute.

Can we get the higher-order benefits of Taylor methods without having to compute complicated derivatives of f?

Yes. Idea:

1. Replace evaluation at f(t_i, w_i) with evaluation at some other point/points f(τ, ω).
2. Instead of w_new = w_old + h f(t_old, w_old) we try w_new = w_old + h f(τ, ω) for some choice of τ and ω.
3. Or even w_new = w_old + h [ a₁ f(τ₁, ω₁) + . . . + a_m f(τ_m, ω_m) ].
4. We choose the points τ_i and ω_i and the coefficients a_i to make this as close as possible to the n^th order Taylor formula.

Since the Taylor formula is complicated, we may not have m = n. That is, you may use m points but not get error O(h^m). Oh well.

In the simplest case when m=1, the formula is called the Midpoint method.
The best time to use is the midpoint between t_old and t_new and the best w-value is the midpoint between w_old and w_new (which is itself approximated by Euler's method). and this will have local truncation error O(h²).
You can see this by expanding w_new in a Taylor series in h.
The terms up to h³ match those of the second-order Taylor method, so cumulatively we have LTE O(h²).

Specifically, the error at each step will look like

---

More generally if we try w_new = w + h [ af(t, w) + bf(t+ch, w+dhf(t, w)) ] then we cannot make it equal to the third-order Taylor method, but if we only try to make it equal to the second-order method, we only get two equations.

There are two popular choices:

1. Trapezoid method ("modified Euler"):
   
2. Heun's method:
   

Of course the local truncation error of all three methods is O(h²).
Notice the right combination of the three methods will eliminate this term and give something with LTE at least O(h³).

But that will involve four points and only give O(h³).