Computational Fluid Dynamics Questions and Answers – Finite Difference Methods – MacCormack’s Technique

This set of Computational Fluid Dynamics Multiple Choice Questions & Answers (MCQs) focuses on “Finite Difference Methods – MacCormack’s Technique”.

1. MacCormack’s technique is __________
a) explicit, finite-difference method
b) implicit, finite-difference method
c) explicit, finite volume method
d) implicit, finite volume method

Explanation: Like the Lax-Wendroff technique, MacCormack’s technique is also particularly suitable for marching solutions of hyperbolic and parabolic partial differential equations. It is also an explicit finite difference scheme for marching solutions.

2. Which series expansion is used by the MacCormack’s technique?
a) Taylor Series
b) Fourier series
c) McLaurin series
d) Laurent series

Explanation: The MacCormack’s technique uses the Taylor series expansion to approximate its time derivatives like the finite difference scheme. But the accuracy here is not dependent on the order of the derivative. It has improved accuracy.

3. What is the order of accuracy of the MacCormack’s technique?
a) Fourth-order
b) Third-order
c) First-order
d) Second-order

Explanation: MacCormack’s technique is second order accurate in both space and time. There is a special method used in MacCormack’s technique to make the order of accuracy two, even after reducing the lengthy algebra.

4. Which of these terms of the Taylor series expansion is used in the MacCormack’s technique?
a) (Δ t)1 and (Δ t)2
b) (Δ t)1
c) (Δ t)0 and (Δ t)1
d) (Δ t)0

Explanation: Only the first two terms in the Taylor series expansion is used in the MacCormack’s technique. The first two terms are (Δ t)0 and (Δ t)1. All other higher-order terms are omitted. But, the order of accuracy is maintained here as two.

5. Expand the term $$\rho_{i,j}^{t+\Delta t}$$ for the MacCormack’s technique.
Note:
t →Current time-step
av → Average time-step between t and t+Δ t.
a) $$\rho_{i,j}^t+(\frac{\partial\rho}{\partial t})_{i,j}^{av}\Delta t$$
b) $$\rho_{i,j}^{av}+(\frac{\partial\rho}{\partial t})_{i,j}^{av} \Delta t$$
c) $$\rho_{i,j}^{av}+(\frac{\partial\rho}{\partial t})_{i,j}^t \Delta t$$
d) $$\rho_{i,j}^t+(\frac{\partial\rho}{\partial t})_{i,j}^t \Delta t$$

Explanation: The MacCormack’s technique uses the previous time-step values of the dependent variable and its derivative is obtained at between times t and t+Δ t. Only the first two terms of the Taylor series expansion is used.
$$\rho_{i,j}^{t+\Delta t} = \rho_{i,j}^t+(\frac{\partial\rho}{\partial t})_{i,j}^{av}\Delta t$$.
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6. Which of these methods is used for finding the average time derivative in MacCormack’s technique?
a) Trial and error method
b) Predictor-corrector method
c) Genetic algorithm
d) Relaxation method

Explanation: To get the time derivative at the average time between t and t+Δ t, the MacCormack’s technique uses the Predictor-corrector method. This is used as we do not know the value for the time derivative at the time-step t+Δ t.

7. How is the value $$(\frac{\partial \rho}{\partial t})_{i,j}^{av}$$ obtained in the MacCormack’s expansion to find $$\rho_{i,j}^{t+\Delta t}$$?
a) Truncated mean of $$(\frac{\partial\rho}{\partial t})_{i,j}^t and (\frac{\partial\rho}{\partial t})_{i,j}^{t+\Delta t}$$
b) Weighted average of $$(\frac{\partial\rho}{\partial t})_{i,j}^t and (\frac{\partial\rho}{\partial t})_{i,j}^{t+\Delta t}$$
c) Geometric mean of $$(\frac{\partial\rho}{\partial t})_{i,j}^t and (\frac{\partial\rho}{\partial t})_{i,j}^{t+\Delta t}$$
d) Arithmetic mean of $$(\frac{\partial\rho}{\partial t})_{i,j}^t and (\frac{\partial\rho}{\partial t})_{i,j}^{t+\Delta t}$$

Explanation: The value of $$(\frac{\partial \rho}{\partial t})_{i,j}^{av}$$ is the arithmetic mean of $$(\frac{\partial \rho}{\partial t})_{i,j}$$ at t and $$(\frac{\partial \rho}{\partial t})_{i,j}$$ at t+Δt.
$$(\frac{\partial \rho}{\partial t})_{i,j}^{av}=\frac{1}{2}[(\frac{\partial\rho}{\partial t})_{i,j}^t+(\frac{\partial\rho}{\partial t})_{i,j}^{t+\Delta t}]$$.

8. Which value is predicted in the predictor step of the MacCormack’s technique?
a) Variable at the average time-step
b) Variable at the upcoming time-step
c) Time derivative of the variable at the upcoming time-step
d) Time derivative of the variable at the average time-step

Explanation: To get the derivative in the upcoming time-step for finding the time derivative of the variable at the average time, the variable at that time step is needed. This is the value which we intend to find using MacCormack’s technique. So, in the predictor step, the value of the variable at that time step is predicted.

9. Which of these values used to find $$(\frac{\partial \rho}{\partial t})_{i,j}^{av}$$ is a predicted one?
a) $$(\frac{\partial\rho}{\partial t})_{i,j}^t$$
b) Neither $$(\frac{\partial\rho}{\partial t})_{i,j}^t nor (\frac{\partial\rho}{\partial t})_{i,j}^{t+\Delta t}$$
c) $$(\frac{\partial\rho}{\partial t})_{i,j}^{t+\Delta t}$$
d) Both $$(\frac{\partial\rho}{\partial t})_{i,j}^t and (\frac{\partial\rho}{\partial t})_{i,j}^{t+\Delta t}$$

Explanation: $$(\frac{\partial\rho}{\partial t})_{i,j}^{t+\Delta t}$$ is predicted using the continuity equation and the value of $$\rho_{i,j}^{t+\Delta t}$$ in the process of finding $$(\frac{\partial \rho}{\partial t})_{i,j}^{av}$$. The continuity equation is used as we need the time rate of change of density.

10. I am using forward differences in the predictor step. Which method would you suggest me to use in the corrector step?
a) Rearward differences
b) Central differences
c) Forward differences
d) Second-order differences

Explanation: If forward differences are used in the predictor step, rearward differences should be used in the corrector step and vice versa. At every time-step, this sequence should be changed while solving a time-marching problem.

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