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

1. Spectral methods are particularly suitable for __________

a) Subsonic flows

b) Boundary layer flows

c) Compressible flows

d) Turbulence modelling

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Explanation: Spectral methods are not so commonly used like the Finite Volume and Finite Difference methods in CFD. But, they are specifically very good methods for analyzing turbulence models, especially with uniform grids.

2. Spectral methods use ___________

a) Fourier series

b) Taylor series

c) McLaurin series

d) Laurent series

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Explanation: Finite Difference methods use Taylor series. In spectral methods, spatial derivatives are evaluated using the Fourier series or one of their generalization. The simplest spectral methods deal with periodic functions specified by their values at a uniformly spaced set of points.

3. What causes aliasing in Spectral methods?

a) Small grid sizes

b) Fourier series

c) Arbitrary values for Fourier series

d) Periodic nature

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Explanation: For Fourier expansion, there is a nearly arbitrary value to be assumed. The results depend on this arbitrary value. If this value is not chosen properly, it will lead to aliasing error. This means pointing the same element more than one time.

4. What is the advantage of using Fourier series in Spectral method?

a) Less grid size

b) Large number of grid points

c) Continuous results

d) Flexibility

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Explanation: The Fourier series can be interpolated to get the dependent function. This will help us to get the results at a continuous space instead of results at particular grid points. This is an advantage over the other discretization techniques.

5. For higher order derivatives, spectral methods ___________

a) can be easily generated

b) are not suitable

c) difficult to generate

d) become invalid

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Explanation: Spectral method can easily be generated for higher derivatives. The Fourier coefficients will vary in higher order derivatives. Other than this, there are not much changes needed for higher orders.

6. Spectral methods are much more accurate than the Finite Difference methods ___________

a) unconditionally

b) conditionally depending on the problem taken

c) conditionally when the number of grid points is small

d) conditionally when grid size is small

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Explanation: The error in the solution decreases exponentially with the number of grid points. The results are more precise when the number of grid points is more. So, the grid size should be small. This makes the spectral method with more grids accurate than the Finite Difference methods.

7. The cost of computing the Fourier coefficients is ___________ (Note: ‘N’ is the number of grid points).

a) N^{3}

b) N^{2}

c) \(\frac{N^2}{2} \)

d) \(\frac{N^3}{3} \)

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Explanation: The computational cost required for computing Fourier coefficients, if done in the most obvious manner, is N

^{2}. This is prohibitively expensive. It is twice that of the backward substitution for Gauss-Elimination method.

8. The cost of computation for Fourier coefficients can be reduced by ___________

a) FFT

b) DFT

c) IDFT

d) IFT

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Explanation: FFT stands for Fast Fourier Transform which is an algorithm for finding Discrete Fourier Transform (DFT) of a sequence or the Inverse Discrete Fourier Transform (IDFT). This is used to reduce the computational cost.

9. What is the cost of computation of FFT? (Note: ‘N’ is the number of grid points).

a) N

b) log N

c) N log N

d) \(\frac{N^2}{2} \)

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Explanation: The cost of computation is reduced by FFT. FFT has a cost of computation of N log N orders. This is much lower than N

^{2}, especially when N is large. This reduces the problem of computation cost in the Spectral method.

10. To make the spectral method advantageous _____________

a) Function must be periodic but grids can be non-uniform

b) Grids should be uniform and function must be periodic

c) Grids should be uniform but function can be non-periodic

d) Grids should be structured and function must be periodic

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Explanation: To get the full advantages of this Spectral method, the function must be periodic of the dependent variable and the grids should be uniform. This makes the Spectral method inflexible when compared to the other discretization methods.

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