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

𝑑
𝑦
𝑑
𝑥
=
𝑦
4
+
3
𝑥
2
+
𝑙
𝑜
𝑔
(
𝑦
)

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.

𝑑
𝑦
𝑑
𝑥
=
𝑦
4
+
3
𝑥
2
+
𝑙
𝑜
𝑔
(
𝑦
)

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 ODE and four global truncation errors corresponding to different numerical methods: Eeuler, Emidpoint, EHeun, and ERK4. We are asked which of the following inequalities is FALSE. Option: ERK4 > EHeun - RK4 (ERK4) is a fourth-order method, meaning its global truncation error scales like O(h^4) for a fixed problem, with a leading constant that depends on the problem and step size h. - Heun’s method (EHeun) is a second-order method, so its global truncation error scales like O(h^2). - For small step sizes h, O(h^4) terms dominate less than O(h^2) terms, and thus the RK4 error is typically much smaller than the Heun error, i.e., ERK4 < EHeun, up to the leading constants. - Therefore, the inequality ERK4 > EHeun would claim RK4 yields a larger error than Heun, which contradicts the standard order-based error behavior. In other words, this statement is generally false for well-behaved problems and common step sizes. Could there be caveats? In practice, the actual comparison depends on the specific problem, step size, and the constants multiplying the leading error terms. It is possible that for unusually large step sizes or pathological problems, the practical observed errors do not strictly follow the asymptotic orders, but the theoretical expectation remains ERK4 < EHeun as h → 0. Since only one option is provided, this is the option whose truth value is incorrect under standard order-aware analysis. The other potential factors, such as stability or implementation details, could influence observed errors but do not overturn the fundamental order-based relationship between fourth-order and second-order methods.

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