This set of Computational Fluid Dynamics Quiz focuses on “Numerical Methods – Multi-grid Approach for Solving Discretized Equations”.

1. The multi-grid approach is used to assist __________

a) iterative solvers

b) direct solvers

c) gradient solvers

d) pre-conditioned solvers

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Explanation: When the iterative solvers are used to solve medium to large system of equations, the error is big and accuracy becomes less. This poses a problem with iterative solvers. So, they are supplemented with the multi-grid approach.

2. Which of these properties are affected when the multi-grid approach is not used?

a) Conservativeness

b) Convergence

c) Consistency

d) Stability

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Explanation: As the accuracy in the iterative solvers for large equations are not good, the rate of convergence is very less. A solution to this problem is given by the multi-grid approach.

3. Which of these errors need a multi-grid approach?

a) Low amplitude error

b) High amplitude error

c) Low frequency error

d) High frequency error

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Explanation: High frequency oscillatory errors are easily eliminated using iterative methods like Jacobi and Gauss-Seidel. But, these techniques cannot handle smooth and low frequency errors without a multi-grid approach.

4. Multi-grid approach switches between ___________ and ____________ grids to meet the errors.

a) structured and unstructured

b) collocated and staggered

c) cylindrical to polar

d) fine and coarse

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Explanation: The low frequency errors create problem in fine grids as the error in one cell is very small and negligible because of the high wavelength of the errors. So, multi-grid approach changes these fine grids into coarser one to make it considerable in one cell.

5. The multi-grid approach is a ___________ process.

a) direct

b) iterative

c) cyclic

d) periodic

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Explanation: The multi-grid approach involves traversal from a fine grid into a coarse one in order to make the error considerable and again another traversal from a coarse grid into a fine one after the error correction.

6. Let r^{k} be the residual in the k^{th} level of multi-grid approach. Which of these give the restriction operator?

a) \(\frac{r^{k+1}}{r^k} \)

b) \(\frac{r^k}{r^{k+1}}\)

c) r^{k+1}-r^{k}

d) r^{k}-r^{k+1}

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Explanation: The first step in the multi-grid approach is the restriction step. Here, the process starts with a fine grid. After a few iterations, the error is transferred to a coarser grid level. Again, some iterations are performed in that step and the process continues. The restriction operator for the error transferred from one step to the higher step is \(\frac{r^{k+1}}{r^k} \).

7. For an algebraic multi-grid approach, the residual in the k^{th} level is __________

a) residual in the (k+1)^{th} level

b) summation of the residuals in the (k)^{th} level

c) residual in the (k-1)^{th} level

d) summation of the residuals in the (k-1)^{th} level

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Explanation: While transferring the errors from one level to another, the residual of the k

^{th}level is given by the summation of the residuals of all the terms in the previous (k-1)

^{th}level.

8. Which of these traversal cycles are possible for an algebraic multi-grid approach?

a) W-cycle

b) V-cycle

c) U-cycle

d) F-cycle

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Explanation: There are three possible traversal cycles for algebraic multi-grid approaches. They are W-cycle, V-cycle and F-cycle. V-cycle is a direct traverse without any nesting. W-cycle involves nesting. F-cycle is a hybrid cycle of W and V-cycles.

9. Errors are transferred from a fine grid to a coarser one. Similarly __________ is transferred from a coarse grid to a finer one.

a) residual

b) correction

c) restriction

d) prolongation

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Explanation: Corrections are transferred from a coarse grid to a finer one. Correction is basically obtained from the solution of the system of equations at the coarse grid. This, in terms of ratios of errors, is transferred to the finer grid.

10. Which of these is the opposite step of restriction?

a) Prolongation

b) Traversal

c) Agglomeration

d) Coarsening

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Explanation: Prolongation is the step where correction is transferred from a coarse grid to a finer one. This is done the same way how errors are transferred from a fine grid to a coarser one in restriction step.

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