Question

What is the correct formula for Runge-Kutta 3rd order method?

• A. \( y = y_0 + \frac{1}{6}(k_1 + 2k_2 + 3k_3) \)
• B. \( y = y_0 + \frac{1}{3}(k_1 + 4k_2 + k_3) \)
• C. \( y = y_0 + \frac{1}{6}(k_1 + 4k_2 + k_3) \)
• D. \( y = y_0 + \frac{1}{6}(k_1 + k_2) \)

Hint:

For higher-order Runge-Kutta methods, the weightings in the formula change depending on the order of the method.

Answer 1

The Correct Option is A

The Runge-Kutta method is used to approximate solutions of ordinary differential equations (ODEs). The 3rd order Runge-Kutta method (RK3) can be expressed by the formula:

\[ y = y_0 + \frac{1}{6}(k_1 + 2k_2 + 3k_3) \]

Where:

• \( k_1 = h f(x_0, y_0) \)
• \( k_2 = h f(x_0 + \frac{h}{2}, y_0 + \frac{k_1}{2}) \)
• \( k_3 = h f(x_0 + h, y_0 - k_1 + 2k_2) \)

Thus, the correct formula for the Runge-Kutta 3rd order method is:

\[ y = y_0 + \frac{1}{6}(k_1 + 2k_2 + 3k_3) \]

This corresponds to option (A).