Question

In the 4th order Runge-Kutta method for solving ordinary differential equations with step size $h<1$, the ratio of the order of local error to the order of global error is

• A. $h$
• B. $h^2$
• C. $\dfrac{1}{h}$
• D. $\dfrac{1}{h^2}$

Hint:

In Runge-Kutta methods, local error is always one power of $h$ higher than the global error. For RK4, local error $\sim h^5$ while global error $\sim h^4$. Thus, the ratio is always $h$.

Answer 1

The Correct Option is A

Step 1: Local Truncation Error (LTE).
- In the 4^th order Runge-Kutta (RK4) method, the local truncation error per step is of the order $\mathcal{O}(h^5)$.
- This means that each step introduces an error proportional to $h^5$.
Step 2: Global Error (GE).
- The global error accumulates over $\frac{1}{h}$ steps (since the total interval is divided into $\frac{1}{h}$ steps of size $h$).
- Therefore, the global error is $\mathcal{O}(h^4)$ because:
\[ \text{Global Error} = \frac{1}{h} \times \mathcal{O}(h^5) = \mathcal{O}(h^4). \] Step 3: Ratio of Orders.
- Order of local error = $h^5$
- Order of global error = $h^4$
- Ratio = $\frac{h^5}{h^4} = h$
Final Answer: $h$