MATH/APPM 4650
Class notes, March 16

Taylor methods

Today's notes will mostly be on the board.

Recall Euler's method says w₀ = y₀,    w_i+1 = w_i + h f(t_i, w_i) should be a good approximation to y(t_i).

It comes from y(t+h) ≈ y(t) + hy'(t) + ... and the fact that we know y'(t) = f(t, y).

If we use more terms in this Taylor series, we should get a better approximation.

But what's y''(t)?

Just use the Chain rule:
For example, if y'(t) = t² + y², then

So instead of Euler's method which would give w_i+1 = w_i + h (t_i² + w_i²), the second-order Taylor method would be w_i+1 = w_i + h (t_i² + w_i²) + h²/2 [2t_i + 2w_i (t_i² + w_i²)].
The error in the second-order Taylor method is the next unused term in the Taylor series, which looks like h³/6 y'''(ξ_i)

This is the error at each step even if the previous step were correct.

The total error, for a given time t, will look like Ch² [exp(Lt) - 1] without roundoff. Here C and L are some numbers depending on derivatives of f and the initial condition.

In general we may use a Taylor method of any order.

1. Start with the general Taylor series formula.
2. Replace y'(t) with f(t, y).
3. Replace y''(t) with f_t(t, y) + f(t, y) f_y(t, y).
4. Continue computing higher derivatives using the differential equation and the chain rule, as needed.
5. Once you have a formula giving y(t+h) only in terms of t and y (not y' or any other derivative), replace t with t_i, replace y(t) with w_i, and replace y(t+h) with w_i+1.

This is the Taylor method of order n.
The error at each step will be O(hⁿ⁺¹), and the overall error will be O(hⁿ).

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Definition: If w_i+1 = H(t_i, w_i) is any method for approximating the solution of y'(t) = f(t, y(t)), then the local truncation error of the method is defined as follows:

1. Assume w_i = y(t_i).
2. By expanding y(t_i + h) in a Taylor series in h, find the error in w_i+1 - y(t_i+h).
3. The local truncation error is then [w_i+1 - y(t_i+h)]/h.

Dividing by h at the end is so that we know not just happens at one step, but rather what the accumulated error is over all n steps.

Usually we only care about the order of the error.
So Euler's method has local truncation error O(h) and the second-order Taylor method has local truncation error O(h²).