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
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?
1 & 0 & 0 \\
2 & 1 & 0 \\
3 & 4 & 1
0 & 0 & 0 \\
2 & 0 & 0 \\
3 & 4 & 0
1 & 0 & 0 \\
2 & 5 & 0 \\
3 & 4 & 6
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3
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 ith iteration, the general form can be given as ______________ (Note: Φi is the value of Φ in the ith iteration).
Explanation: The global matrix is
Replacing A with M-N,
At a particular step, we only know the Φ values of the previous step only. So,
4. Consider the global matrix AΦ=b. What is the residual at the ith iteration? (Note: Φi is the value of Φ in the ith iteration).
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 AΦi-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
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.
c) spectral properties
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?
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
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
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).
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.
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