Question textMatch the local and global truncation errors for the following ODE solving methods: Euler's method: Answer 1 Question 23[select: , local: O(h), global: O(h^2), local: O(h^2), global: O(h), local:(O(h^2), global: O(h^3), local: O(h^3), global: O(h^2), local: O(h^3), global: O(h^4)] Heun's method: Answer 2 Question 23[select: , local: O(h), global: O(h^2), local: O(h^2), global: O(h), local:(O(h^2), global: O(h^3), local: O(h^3), global: O(h^2), local: O(h^3), global: O(h^4)] Midpoint method: Answer 3 Question 23[select: , local: O(h), global: O(h^2), local: O(h^2), global: O(h), local:(O(h^2), global: O(h^3), local: O(h^3), global: O(h^2), local: O(h^3), global: O(h^4)]

Question

BlackTom AI · Accepted Answer

We are asked to match local and global truncation errors for three ODE solving methods: Euler's method, Heun's method, and the Midpoint method.

Option by option reasoning:

- Euler's method:
  - Local truncation error measures the error per step when approximating the solution over one step, assuming the previous value is exact. For Euler's method, this local error scales with h^2, i.e., local ~ O(h^2).
  - Global truncation error accumulates over all steps and scales linearly with the step size h, i.e., global ~ O(h).
  - Therefore, the correct pairing is local: O(h^2), global: O(h).

- Heun's method:
  - Heun's method (a predictor-corrector / improved Euler) is a second-order method. Its local truncation error is of order h^3, i.e., local ~ O(h^3).
  - Its global truncation error is of order h^2, i.e., global ~ O(h^2).
  - Thus, the correct pairing is local: O(h^3), global: O(h^2).

- Midpoint method:
  - The classical (modified Euler) midpoint method is also a second-order method. Its local truncation error scales as h^3, i.e., local ~ O(h^3).
  - Its global truncation error scales as h^2, i.e., global ~ O(h^2).
  - Hence, the correct pairing is local: O(h^3), global: O(h^2).

Putting these together, each method has its respective local/global pair as described above, matching the commonly known orders for these Runge–Kutta-like schemes.

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