Consider the following ODE, which is solved using a numerical method.

Let

Eeuler be the global truncation error of the Forward Euler method when solving the equation above
Emidpoint be the global truncation error of the midpoint method when solving the equation above
EHeun be the global truncation error of the Heun method when solving the equation above
ERK4 be the global truncation error of the RK4 method when solving the equation above

Which of the following inequalities is FALSE?

Question

Consider the following ODE, which is solved using a numerical method.

Let

Eeuler be the global truncation error of the Forward Euler method when solving the equation above
Emidpoint be the global truncation error of the midpoint method when solving the equation above
EHeun be the global truncation error of the Heun method when solving the equation above
ERK4 be the global truncation error of the RK4 method when solving the equation above

Which of the following inequalities is FALSE?

BlackTom AI · Accepted Answer

The question presents an ordinary differential equation solved numerically, and defines four truncation errors for different methods: 
Eeuler for Forward Euler, Emidpoint for the midpoint method, EHeun for Heun's method, and ERK4 for the classical fourth-order Runge–Kutta method. The task asks which inequality is FALSE.
Option: ERK4 > EHeun

To assess this, recall key numerical analysis facts about these methods:
- The RK4 method has local truncation error of order h^5 and global (accumulated) truncation error of order h^4, meaning its error decreases proportionally to h^4 as the step size h becomes small.
- Heun’s method is a second-order Runge–Kutta method, with global truncation error of order h^3 (or more commonly described as order 2 in some conventions, but the standard widely accepted global error order is 2 for RK methods like Heun when considering the typical definitions). This means its error scales roughly as h^2 or h^3 depending on the precise definition, but in any standard viewing RK4 outperforms RK2 methods for small h.
- In contrast, Forward Euler is first-order with global error about O(h), which is larger than the errors of RK2 and RK4 for sufficiently small h.

Given these orders, for sufficiently small step size h, the magnitude of the global truncation error satisfies roughly ERK4 ∝ h^4 and EHeun ∝ h^2 (or at worst h^3 in some detailed treatments), so ERK4 is smaller in magnitude than EHeun when h is small. Therefore the inequality ERK4 > EHeun would be false in that regime.

Common misconceptions to watch out for:
- If one imagines only local truncation error without considering how errors accumulate, one might misjudge the comparison. The global (accumulated) error is what matters in a fixed interval, and RK4’s higher order ensures a smaller global error for the same h.
- It is not correct to assume that a higher-order method has a larger error; higher-order methods typically yield smaller errors as h decreases, assuming the problem is well-behaved and h is in a reasonable range.

In summary, based on the standard order of these methods, RK4’s global error is of higher order (smaller for small h) than Heun’s, so the stated inequality ERK4 > EHeun does not hold under typical conditions. The other possible considerations would involve pathological step sizes or highly stiff problems, but under ordinary circumstances the comparison described above holds.

BN5206 All Canvas questions [NON-GRADED, FOR PRACTICE]

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