This set of Computational Fluid Dynamics Multiple Choice Questions & Answers (MCQs) focuses on “Numerical Methods – Iterative Solvers for Discretized Equations”.

1. Which of these statements is incorrect about iterative methods?

a) Low computational cost

b) Low computer storage

c) Not suitable for sparse matrices

d) Needs initial guess

View Answer

Explanation: Iterative methods are chosen when the coefficient matrix is sparse (less non-zero elements). They need less storage and less computational cost. They start the solution from an initial guess and proceed to find the answer from this initial guess.

2. Let us divide the coefficient matrix into this form: A=D+L+U. Which of these matrices can be L in this equation?

a) \(\begin{bmatrix}

1 & 0 & 0 \\

2 & 1 & 0 \\

3 & 4 & 1

\end{bmatrix}\)

b) \(\begin{bmatrix}

0 & 0 & 0 \\

2 & 0 & 0 \\

3 & 4 & 0

\end{bmatrix}\)

c) \(\begin{bmatrix}

1 & 0 & 0 \\

2 & 5 & 0 \\

3 & 4 & 6

\end{bmatrix}\)

d) \(\begin{bmatrix}

1 & 0 & 0 \\

0 & 2 & 0 \\

0 & 0 & 3

\end{bmatrix}\)

View Answer

Explanation: In A=D+L+U, A is the coefficient matrix; D is a diagonal matrix; L is a strict lower triangular matrix; U is a strict upper triangular matrix. A strict lower triangular matrix contains non-zero elements only below its main diagonal.

3. Consider the global matrix AΦ=b. Let the coefficient matrix A=M-N. At the i^{th} iteration, the general form can be given as ______________ (Note: Φ^{i} is the value of Φ in the i^{th} iteration).

a) MΦ^{i}=NΦ^{i-1}+b

b) MΦ^{i}=NΦ^{i}+b

c) MΦ^{i}=NΦ^{i-1}-b

d) MΦ^{i}=NΦ^{i}-b

View Answer

Explanation: The global matrix is

AΦ=b

Replacing A with M-N,

(M-N)Φ=b

MΦ=NΦ+b

At a particular step, we only know the Φ values of the previous step only. So,

MΦ

^{i}=NΦ

^{i-1}+b.

4. Consider the global matrix AΦ=b. What is the residual at the ith iteration? (Note: Φ_{i} is the value of Φ in the i^{th} iteration).

a) AΦ^{i-1}+b

b) AΦ^{i-1}-b

c) AΦ^{i}+b

d) AΦ^{i}-b

View Answer

Explanation: While solving the system AΦ=b, AΦ-b should be equal to zero. Since, the iterative method will not produce such an answer, the decision when to stop the iteration relies upon a tolerance value. When the residual AΦ

^{i}-b becomes less than the tolerance, iterations are stopped.

5. When compared to the Jacobi method, the Gauss-Siedel method ____________

a) has better convergence and needs less memory

b) has less convergence and needs more memory

c) has less convergence and needs less memory

d) has better convergence and needs more memory

View Answer

Explanation: Gauss-Seidel method uses the latest values at a particular iteration. So, it has better convergence. The same way, as there is no need for storing the values of previous iterations, they require less memory too.

6. Preconditioners improve ____________ of the coefficient matrix.

a) sparsity

b) rank

c) spectral properties

d) Singularity

View Answer

Explanation: For a system of equations to have a better rate of convergence, the coefficient matrix should have a less spectral radius. So, Preconditioners are used to improve the spectral characteristics of a system to give the same solution.

7. What does the letter ‘I’ stand for in ILU decomposition problem?

a) Inbuilt

b) Incomplete

c) Iterative

d) Imbalanced

View Answer

Explanation: ILU means Incomplete LU decomposition method. This does incomplete factorization of the coefficient matrix into upper and lower triangular matrices. The L and U matrices have the same number of non-zero elements as in the lower and upper parts of A.

8. In which of these methods, after factorization, the pattern of zero elements in the combined L and U matrices the same as the original coefficient matrix?

a) LU decomposition

b) ILU decomposition

c) ILU(0) decomposition

d) ILU(1) decomposition

View Answer

Explanation: The extra non-zero elements in the factorized matrices are called fill-ins. In ILU(p), p indicates the order of fill-in allowed. So, where there is no fill-in, the pattern of combined L and U matrices and the coefficient matrix will be the same.

9. Which of these methods is not restricted to symmetric positive definite matrices?

a) The method of steepest descent

b) Conjugate gradient method

c) Bi-conjugate gradient method

d) Gauss-Seidel method

View Answer

Explanation: The gradient iterative solvers are restricted to symmetric positive definite matrices. The gradient methods are:

- The method of Steepest descent
- Conjugate gradient method
- Bi-conjugate gradient method
- Preconditioned bi-conjugate gradient method.

10. In incomplete Cholesky decomposition, the preconditioner matrix produced is ____________ (Note: L is the factorized lower triangular matrix and U is the factorized lower triangular matrix).

a) UL’

b) LU’

c) LL’

d) LL^{2}

View Answer

Explanation: In incomplete Cholesky decomposition, factorization is done only for the lower triangular matrix and the preconditioner matrix is LL’. The coefficient matrix is also approximately equal to the preconditioner matrix.

**Sanfoundry Global Education & Learning Series – Computational Fluid Dynamics.**

To practice all areas of Computational Fluid Dynamics, __here is complete set of 1000+ Multiple Choice Questions and Answers__.

**If you find a mistake in question / option / answer, kindly take a screenshot and email to [email protected]**

**Related Posts:**

- Check Computational Fluid Dynamics Books
- Check Chemical Engineering Books
- Apply for Mechanical Engineering Internship
- Practice Mechanical Engineering MCQs
- Apply for Chemical Engineering Internship