This set of Computational Fluid Dynamics Multiple Choice Questions & Answers (MCQs) focuses on “Transient Flows – Runge Kutta Method”.
1. The second-order Runge-Kutta method uses __________ as a predictor.
a) backward order method
b) forward Euler method
c) midpoint rule
d) multipoint method
Explanation: The second-order Runge-Kutta method includes two steps. The first step can be called a half-step predictor. This is based on the forward Euler method which is an explicit method of first-order accuracy.
2. Which of these correctors does the second-order Runge-Kutta method use?
a) Backward Euler corrector
b) Forward Euler corrector
c) Trapezoidal corrector
d) Midpoint rule corrector
Explanation: The second step of the second-order Runge-Kutta method is the corrector step. For this correction, midpoint rule is used. This step makes this Runge-Kutta method a second-order method.
3. How many steps does the fourth-order Runge-Kutta method use?
a) Two steps
b) Five steps
c) Four steps
d) Three steps
Explanation: All the Runge-Kutta methods are of high orders. The fourth-order Runge-Kutta method is a method which uses four steps. These four steps include the predictor and the corrector steps.
4. The first two steps of the fourth-order Runge-Kutta method finds the value at which point?
a) At the (n+0.5)th point
b) At the (n+1)th point
c) At the (n-1)th point
d) At the nth point
Explanation: The first two steps of the fourth-order Runge-Kutta method find the values at the (n+0.5)th point. It does not directly move to the next step. It finds the value at an intermediate point between the current and the next points.
5. How many predictor and corrector steps does the fourth-order Runge-Kutta method use?
a) Three predictor and one corrector steps
b) One predictor and three corrector steps
c) Two predictor and two corrector steps
d) One predictor and two corrector steps
Explanation: The fourth-order Runge-Kutta method totally has four steps. Among these four steps, the first two are the predictor steps and the last two are the corrector steps. All these steps use various lower order methods for approximations.
6. The first two steps of the fourth-order Runge-Kutta method use __________
a) Euler methods
b) Forward Euler method
c) Backward Euler method
d) Explicit Euler method
Explanation: All the steps of the Runge-Kutta method use the two-level formulae for initial value problems. The first step uses the forward Euler method and the second step uses the backward Euler method. Collectively, we can say that these two steps use Euler methods.
7. The final corrector of the fourth-order Runge-Kutta method uses ___________
a) Midpoint rule
b) Backward Euler method
c) Simpson’s rule
d) Trapezoidal rule
Explanation: The third step of the fourth-order Runge-Kutta method uses midpoint rule to correct the values and the last step uses Simpson’s rule. This renders a fourth-order accuracy to the Runge-Kutta method.
8. Consider an nth 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
Explanation: At each step of the Runge-Kutta method, the derivate has to be evaluated n times. Here, ‘n’ is the order of accuracy of the Runge-Kutta method. This is a major disadvantage of Runge-Kutta methods.
9. Which of these statements is correct?
a) When the order of accuracy is the same for two methods, the accuracy is also the same
b) Runge-Kutta method interpolates at more than one point in a time interval
c) Runge-Kutta method is not a multipoint method
d) An nth order Runge-Kutta method is more accurate than the nth order multipoint method
Explanation: When comparing the Runge-Kutta method and the multipoint method, even if the order of accuracy is the same, the Runge-Kutta method is more accurate. This is because the coefficient of the Runge-Kutta method is small.
10. Which of these is a disadvantage of the Runge-Kutta method over the multipoint method?
a) Computational stability
b) Computational cost
Explanation: Though the Runge-Kutta methods are advantageous in the accuracy and stability perspectives, this comes at the cost of computational cost. Computationally, the Runge-Kutta methods are very expensive when compared to the other methods.
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