Question

For the initial value problem \[ \frac{dy}{dx} = f(x, y), \quad y(x_0) = y_0, \] generate approximations \( y_n \) to \( y(x_n) \) using the recursion formula \[ y_n = y_{n-1} + a k_1 + b k_2, \] where \[ k_1 = h f(x_{n-1}, y_{n-1}), \quad k_2 = h f(x_{n-1} + \beta h, y_{n-1} + \beta k_1). \] Which one of the following choices of \( a, b, \alpha, \beta \) gives the Runge-Kutta method of order 2?

• A. \( a = 1, b = 1, \alpha = 0.5, \beta = 0.5 \)
• B. \( a = 0.5, b = 0.5, \alpha = 2, \beta = 2 \)
• C. \( a = 0.25, b = 0.75, \alpha = 2/3, \beta = 2/3 \)
• D. \( a = 0.5, b = 0.5, \alpha = 1, \beta = 2 \)

Hint:

For Runge-Kutta methods, check the consistency and accuracy conditions for the given coefficients.

Answer 1

The Correct Option is C

Step 1: Understanding the Runge-Kutta method. The Runge-Kutta method of order 2 satisfies: \[ y_n = y_{n-1} + a k_1 + b k_2, \] where \( k_1 \) and \( k_2 \) involve weighted evaluations of \( f(x, y) \). The coefficients \( a, b, \alpha, \beta \) determine the order and accuracy of the method. Step 2: Verifying the choice. For \( a = 0.25, b = 0.75, \alpha = 2/3, \beta = 2/3 \), the method satisfies the conditions for the second-order accuracy: \[ a + b = 1, \quad b \cdot \beta = \frac{1}{2}. \] Step 3: Conclusion. The correct choice of coefficients is \( {(3)} \): \( a = 0.25, b = 0.75, \alpha = 2/3, \beta = 2/3 \).