Which of the following statements regarding numerical methods for initial value problems $y' = f(x, y)$ is/are TRUE?
Options:
(A) The forward Euler method is an explicit single-step method with local truncation error $\mathcal{O}(h^2)$ and global truncation error $\mathcal{O}(h)$
(B) The backward Euler method is an implicit single-step method and is unconditionally A-stable for linear test equations $y' = \lambda y$ with $\operatorname{Re}(\lambda) < 0$
(C) Multistep methods like Adams–Bashforth require starter values from single-step methods like Runge–Kutta
(D) Runge–Kutta methods are self-starting and do not require prior historical steps beyond $(x_n, y_n)$
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
1. **Euler Truncation Error (Option A)**: Taylor expansion shows local error is $\frac{h^2}{2} y''(\xi) = \mathcal{O}(h^2)$, which accumulates over $N = (b-a)/h$ steps to a global error of $\mathcal{O}(h)$ (True).
2. **A-Stability of Backward Euler (Option B)**: The amplification factor $R(z) = \frac{1}{1 - z}$ satisfies $|R(z)| < 1$ for all $\operatorname{Re}(z) < 0$, making it unconditionally A-stable (True).
3. **Multistep Starters (Option C)**: An $s$-step method requires $s$ previous points ($y_n, y_{n-1}, \dots, y_{n-s+1}$), which cannot be provided solely by initial condition $y_0$ and thus require Runge–Kutta starters (True).
4. **Self-starting Single-Step Nature (Option D)**: All Runge–Kutta methods compute intermediate slopes strictly inside $(x_n, x_{n+1})$, needing only $(x_n, y_n)$ (True).