Question

Consider an n^th order accurate Runge-Kutta method. How many times is the derivative evaluated at the fourth time-step?

(a) one time

(b) two times

(c) four times

(d) n times

This question was posed to me during an interview for a job.

The question is from Transient Flows in portion Transient Flows of Computational Fluid Dynamics

Answer 1

The correct answer is:

(c) four times

Explanation:

In an $n$-th order accurate Runge-Kutta method, the derivative (or the function that describes the system) is evaluated multiple times at each time step. The number of times the derivative is evaluated depends on the specific method used. For standard Runge-Kutta methods (like the classical 4th-order method), the derivative is evaluated at each of the intermediate stages (4 stages for 4th-order Runge-Kutta).

For a 4th-order Runge-Kutta method, the derivative is evaluated 4 times per time step: once for each of the intermediate stages k1,k2,k3,k4k_1, k_2, k_3, k_4k1​,k2​,k3​,k4​.

Since the question asks about the 4th time-step, in a general $n$-th order accurate method, the derivative would typically be evaluated at each intermediate stage, and for an $n=4$ case, this happens 4 times at each time step.

Thus, the number of evaluations of the derivative is four times.

Correct Answer: (c) four times