Computational Fluid Dynamics Questions and Answers – Discretization Aspects – Thomas Algorithm

This set of Computational Fluid Dynamics Multiple Choice Questions & Answers (MCQs) focuses on “Discretization Aspects – Thomas Algorithm”.

1. Thomas algorithm is a ___________
a) Linear equations solver
c) Discretization method
d) Linear least square system

Explanation: Using a discretization method, the governing partial differential equation are converted into a system of algebraic equations. These discretized equations are solved using the Thomas algorithm.

2. Thomas algorithm can be used to solve __________
a) any matrix
b) all square matrices
c) only penta-diagonal matrices
d) only tri-diagonal matrices

Explanation: The other name of the Thomas algorithm is Tri-diagonal matrix algorithm. Tri-diagonal matrices are matrices with non-zero elements in the main diagonal and the diagonals above and below it.

3. Thomas algorithm is _________
a) an analytical method
b) a direct method
c) an iterative method
d) a least squares method

Explanation: Thomas algorithm solves a system of equations with non-repeated sequence of operations. It is a direct method to solve the system without involving repeated iterations and converging solutions.

4. Consider a system of equations where the ith equation is ai Φi=bi Φ(i+1)+ci Φ(i+1)+di. While solving this system using Thomas algorithm, we get Φi=Pi Φ(i+1)+Qi. What are Pi and Qi?
a) $$P_i=\frac{c_i Q_{i-1}+d_i}{a_i-c_i P_{i-1}};Q_i=\frac{b_i}{a_i-c_i P_{i-1}}$$
b) $$P_i=\frac{b_i}{a_i-c_i P_{i-1}};Q_i=\frac{c_i Q_{i-1}+d_i}{a_i-c_i P_{i-1}}$$
c) $$P_i=\frac{c_i Q_{i-1}+b_i}{a_i-c_i P_{i-1}};Q_i=\frac{d_i}{a_i-c_i P_{i-1}}$$
d) $$P_i=\frac{d_i}{a_i-c_i P_{i-1}};Q_i=\frac{c_i Q_{i-1}+b_i}{a_i-c_i P_{i-1}}$$

Explanation: As given,
Φi = PiΦi+1+Qi
Φi-1 = Pi-1Φi+Qi-1
The ith equation is,
aiΦi = bi Φi+1 + ciΦi-1 + di
aiΦi = bi Φi+1 + ci(Pi-1Φi + Qi-1) + di
aiΦi – ciPi-1Φi = bi Φi+1+ciQi-1+di
Φi(ai-ci Pi-1) = biΦi+1+ci Qi-1+di
$$\Phi_i = \frac{b_i}{a_i-c_i P_{i-1}}\Phi_{i+1} + \frac{c_i Q_{i-1}+d_i}{a_i-c_i P_{i-1}}$$
Therefore,
$$P_i = \frac{b_i}{a_i-c_i P_{i-1}};Q_i=\frac{C_i Q_{i-1}+d_i}{a_i-c_i P_{i-1}}$$.

5. Let the ith equation of a system of n equations be aiΦi=bi Φi+1+ciΦi-1+di. Which of these is correct?
a) cN=0; bN=0
b) cN=0; b1=0
c) c1=0; bN=0
d) c1=0; b1=0

Explanation: Node 1 will not have a previous node (c1=0). The last node will not have the next node (bN=0).
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6. Using the Thomas algorithm, if the ith unknown is Φi=Pi Φi+1+Qi. what is the last unknown value ΦN equal to?
a) 0
b) PN
c) QN
d) 1

Explanation: For the last equation, bN=0. So, PN=0. ΦN=PNΦN+1+QN=QN.

7. While solving a system of equations with the Thomas algorithm, in which order are the values of Pi and Qi found?
a) Backwards
b) Forward
c) Simultaneously
d) Depends on the problem

Explanation: To find the current values of Pi and Qi, the previous values Pi-1 and Qi-1 should be known. So, this is started from the first equation and done in forward order.

8. After finding all the values of Pi and Qi, in which order are the values of Φi found?
a) Forward
b) Simultaneously
c) Backwards
d) Depends on the problem

Explanation: The last value of ΦN can be found using QN. Using this, the previous value is found using the formula Φi=PiΦi+1+Qi. So, it is done backwards.

9. Consider a system of equations where the ith equation is aiΦi=bi Φi+1+ciΦi-1+di. While solving this system using Thomas algorithm, we get Φi=PiΦi+1+Qi. What are P1 and Q1?
a) $$P_1=\frac{d_1}{a_1-c_1};Q_1=\frac{b_1}{a_1-c_1}$$
b) $$P_1=\frac{b_1}{a_1-c_1};Q_1=\frac{d_1}{a_1-c_1}$$
c) $$P_1=\frac{d_1}{a_1};Q_1=\frac{b_1}{a_1}$$
d) $$P_1=\frac{b_1}{a_1};Q_1=\frac{d_1}{a_1}$$

Explanation: In general,
$$P_i=\frac{b_i}{a_i-c_i P_{i-1}};Q_i=\frac{c_i Q_{i-1}+d_i}{a_i-c_i P_{i-1}}$$
As c1=0,
$$P_1=\frac{b_1}{a_1};Q_1=\frac{d_1}{a_1}$$.

10. A system of equations with which of these coefficient matrices can be solved using the Thomas algorithm?
a) $$\begin{bmatrix} 1 & 2 & 0 \\ 3 & 4 & 5 \\ 0 & 6 &7 \end{bmatrix}$$
b) $$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 7 \end{bmatrix}$$
c) $$\begin{bmatrix} 1 & 0 & 0 \\ 3 & 4 & 0 \\ 5 & 6 & 7 \end{bmatrix}$$
d) $$\begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 5 & 6 & 7 \end{bmatrix}$$

Explanation: Thomas algorithm can be used to solve tri-diagonal matrices only.
$$\begin{bmatrix} 1 & 2 & 0 \\ 3 & 4 & 5 \\ 0 & 6 &7 \end{bmatrix}$$is a tri-diagonal matrix. $$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 7 \end{bmatrix}$$ is an upper triangular matrix.$$\begin{bmatrix} 1 & 0 & 0 \\ 3 & 4 & 0 \\ 5 & 6 & 7 \end{bmatrix}$$ is a lower triangular matrix. $$\begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 5 & 6 & 7 \end{bmatrix}$$ is a square matrix.

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