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
Show Hint
- $h$
- $h^2$
- $\dfrac{1}{h}$
- $\dfrac{1}{h^2}$
The Correct Option is A
Solution and Explanation
- In the 4th 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: $h$
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Identify the option that has the most appropriate sequence such that a coherent paragraph is formed:
Statement:
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