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

This set of Computational Fluid Dynamics Interview Questions and Answers for Experienced people focuses on “Numerical Methods – Direct Solvers for Discretized Equations”.

1. 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

Explanation: 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. 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

Explanation: 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. 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

Explanation: 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. 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) $$\Phi_i=\frac{b_i-\sum_{j=i+1}^N a_{ij}\phi_j}{A_{ii}}$$
b) $$\Phi_i=\frac{b_i-A_{ij}\phi_j}{A_{ii}}$$
c) $$\Phi_i=\frac{\sum_{j=i+1}^NA_{ij}\phi_j}{A_{ii}}$$
d) $$\Phi_i=b_i-\frac{\sum_{j=i+1}^NA_{ij}\phi_j}{A_{ii}}$$

Explanation: The system is solved backwards from the last step. Using the previous value of φ, the current value is found. The formula is
$$\Phi_i=\frac{b_i-\sum_{j=i+1}^N a_{ij}\phi_j}{A_{ii}}$$.

5. If N is the number of unknowns, the number of operations required for backward substitution is ____________
a) $$\frac{N^3}{3}$$
b) $$\frac{N^3}{2}$$
c) $$\frac{N^2}{2}$$
d) $$\frac{N^2}{3}$$

Explanation: The number of operations required for the overall Gauss elimination method is $$\frac{N^3}{3}$$. The number of operations for backward substitution is $$\frac{N^2}{2}$$. This high computational cost is the disadvantage of the Gauss elimination method.
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6. Which of these is true for the LU decomposition method?
a) LΦ=Ub
b) LUΦ=b
c) Φ=LUb
d) UΦ=Lb

Explanation: 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. 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

Explanation: LU decomposition is computationally more expensive than the Gauss elimination method. The number of operations involved is $$\frac{2N^3}{3}$$. This is because the same process of Gauss elimination is carried out twice in the LU decomposition case.

8. 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

Explanation: 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. TDMA and PDMA are particularly suitable for _____________
a) Cartesian grid problems
b) Curvilinear grid problems
c) Unstructured grid problems
d) Structured grid problems

Explanation: 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. 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

Explanation: 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.

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