Which of the following statements is/are TRUE regarding Runge–Kutta methods for solving $\frac{dy}{dx} = f(x, y)$?
Options:
(A) Runge–Kutta methods are single-step methods that do not require prior historical steps
(B) The second-order Runge–Kutta method (RK2) is equivalent to Heun's (Modified Euler) method
(C) The classical RK4 method requires four function evaluations per time step
(D) The standard first-order Runge–Kutta method (RK1) is identical to Euler's forward method
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
1. Unlike multistep methods (e.g. Adams–Bashforth), Runge–Kutta methods compute $y_{n+1}$ using only $(x_n, y_n)$, so they are self-starting single-step methods (A is true).
2. RK2 with weights $k_1 = h f(x_n, y_n)$ and $k_2 = h f(x_n + h, y_n + k_1)$ and $y_{n+1} = y_n + \frac{1}{2}(k_1 + k_2)$ is precisely Heun's method (B is true).
3. Classical RK4 computes four slopes $k_1, k_2, k_3, k_4$, requiring 4 evaluations of $f(x, y)$ per step (C is true).
4. RK1 evaluates $k_1 = h f(x_n, y_n)$ with $y_{n+1} = y_n + k_1$, which is Euler's forward method (D is true).