Consider the Runge-Kutta method with the Butcher tableau
What kind of nonlinear equation(s) does this method solve in every step (i.e. when computing yi+1 from yi) when implemented in a smart way (see e.g. Remark 9.16)?
a. none (All values can be computed explicitly. There is no need to solve any equation.)
b. one equation in n1
c. two equations, one in n₁, and one in n2
d. two coupled equations, each of them in both n₁ and 72
e. one equation in n2
The Correct Answer and Explanation is :
The correct answer is:
d. Two coupled equations, each of them in both ( n_1 ) and ( n_2 ).
Explanation:
The given Butcher tableau corresponds to a two-stage Runge-Kutta method. The method involves solving intermediate stages ( k_1 ) and ( k_2 ), which are computed as:
[
k_1 = f\left( t_i + c_1 h, y_i + h \sum_{j=1}^{s} a_{1j} k_j \right)
]
[
k_2 = f\left( t_i + c_2 h, y_i + h \sum_{j=1}^{s} a_{2j} k_j \right)
]
where ( c_1, c_2 ) and the coefficients ( a_{ij} ) are taken from the Butcher tableau.
In this case, the Butcher tableau shows that the method is implicit, meaning that ( k_1 ) and ( k_2 ) appear on both the left and right sides of the equations. This results in a nonlinear system where both ( k_1 ) and ( k_2 ) must be solved simultaneously.
Since the function ( f ) is generally nonlinear, the equations for ( k_1 ) and ( k_2 ) must be solved using a root-finding method, such as Newton’s method. The system consists of two coupled nonlinear equations because each equation depends on both ( k_1 ) and ( k_2 ). Thus, they cannot be solved independently.
This is different from explicit Runge-Kutta methods, where each stage can be computed explicitly without solving equations.
Hence, the method requires solving two coupled nonlinear equations at each step, making answer (d) the correct choice.
Here is the image of the Runge-Kutta Butcher tableau as requested. Let me know if you need any modifications or further explanations!
