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# Computational Fluid Dynamics – Numerical Methods – Direct Solvers for Discretized Equations

1 - Question

The coefficient matrix in the global matrix format of the algebraic equations is a ______________

a) sparse matrix

b) dense matrix

c) diagonal matrix

d) symmetric matrix

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Answer: aExplanation: Each row in the coefficient matrix represents an equation corresponding to a single node. This equation contains coefficients of the neighbouring elements only depending on the element connectivity of the owner element. So, all other elements of that row become zero. This makes the coefficient matrix sparse.

2 - Question

Which of these direct methods is suitable only for banded matrices?

a) Gauss elimination

b) LU decomposition

c) PDMA

d) LU decomposition by Gauss elimination

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Answer: cExplanation: PDMA stands for Penta-Diagonal Matrix Algorithm. This is used for solving a Penta-diagonal matrix which has non-zero elements only in its main diagonal and two diagonals above and below it. A Penta-diagonal matrix is a sparse (fewer non-zero elements) banded matrix.

3 - Question

After the forward elimination step of the Gauss elimination method, the coefficient matrix is reduced to ____________

a) a lower triangular matrix

b) an upper triangular matrix

c) a diagonal matrix

d) a banded matrix

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Answer: bExplanation: After the forward elimination step, the main diagonal elements and the elements above it are non-zeros. All the elements below the main diagonal are zeros. So, we can say that it gives an upper triangular matrix to be solved.

4 - Question

Which of these formulae is used in the backward substitution step of Gauss-elimination method?

Note: The global matrix is

AΦ=b

i → row number.

j → column number.

N→ Number of unknowns.

a) Φi=bi−∑Nj=i+1aijϕjAii

b) Φi=bi−AijϕjAii

c) Φi=∑Nj=i+1AijϕjAii

d) Φi=bi−∑Nj=i+1AijϕjAii

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Answer: aExplanation: The system is solved backwards from the last step. Using the previous value of φ, the current value is found. The formula is Φi=bi−∑Nj=i+1aijϕjAii.

5 - Question

If N is the number of unknowns, the number of operations required for backward substitution is ____________

a) N33

b) N32

c) N22

d) N23

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Answer: cExplanation: The number of operations required for the overall Gauss elimination method is N33. The number of operations for backward substitution is N22. This high computational cost is the disadvantage of the Gauss elimination method.

6 - Question

Which of these is true for the LU decomposition method?

a) LΦ=Ub

b) LUΦ=b

c) Φ=LUb

d) UΦ=Lb

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Answer: bExplanation: For the LU decomposition method, A=LU Where L and U stand for Lower and Upper triangular matrices respectively. Substituting in the global matrix, LUΦ=b.

7 - Question

The number of operations for LU decomposition method is ____________ the number of operations for the Gauss elimination method.

a) twice

b) half of

c) one-third of

d) thrice

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Answer: aExplanation: LU decomposition is computationally more expensive than the Gauss elimination method. The number of operations involved is 2N33. This is because the same process of Gauss elimination is carried out twice in the LU decomposition case.

8 - Question

Consider the global matrix AΦ=b. If my coefficient matrix A is the same for different b vectors, which of these methods is economic?

a) Gauss elimination

b) TDMA

c) LU decomposition

d) PDMA

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Answer: cExplanation: Once matrix A is factorized in the LU decomposition method, the same factorized matrices can be used to solve different b vectors. Because decomposition does not depend upon the b vector. This is the major advantage of the LU decomposition method

9 - Question

TDMA and PDMA are particularly suitable for _____________

a) Cartesian grid problems

b) Curvilinear grid problems

c) Unstructured grid problems

d) Structured grid problems

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Answer: dExplanation: When a structured grid is used for discretization, it results in a coefficient matrix with its non-zero elements aligning along a few diagonals. The number of non-zero diagonals depends on the discretization stencil and the dimension of the problem. So, TDMA and PDMA are suitable to solve this kind of banded matrix.

10 - Question

The general equation for PDMA is aiΦi+biΦi+2+ci Φi+1+diΦi-1+eiΦi-2=fi. Which of the following is incorrect? (Note: ‘N’ is the number of unknowns).

a) e2=0

b) cN-1=0

c) e1=0

d) cN=0

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Answer: bExplanation: By analysing the equation for PDMA, for the first two equations, d1=e1=e2=0 Similarly, for the last two equations, bN-1=bN=cN=0.