Engineering Questions with Answers - Multiple Choice Questions

# Computational Fluid Dynamics Q – Discretization Aspects – Thomas Algorithm

1 - Question

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 - Question

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 - Question

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 - Question

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) Pi=ciQi−1+diai−ciPi−1;Qi=biai−ciPi−1
b) Pi=biai−ciPi−1;Qi=ciQi−1+diai−ciPi−1
c) Pi=ciQi−1+biai−ciPi−1;Qi=diai−ciPi−1
d) Pi=diai−ciPi−1;Qi=ciQi−1+biai−ciPi−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 Φi=biai−ciPi−1Φi+1+ciQi−1+diai−ciPi−1 Therefore, Pi=biai−ciPi−1;Qi=CiQi−1+diai−ciPi−1.

5 - Question

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).

6 - Question

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 - Question

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 - Question

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 - Question

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) P1=d1a1−c1;Q1=b1a1−c1
b) P1=b1a1−c1;Q1=d1a1−c1
c) P1=d1a1;Q1=b1a1
d) P1=b1a1;Q1=d1a1
Explanation: In general, Pi=biai−ciPi−1;Qi=ciQi−1+diai−ciPi−1 As c1=0, P1=b1a1;Q1=d1a1.

10 - Question

A system of equations with which of these coefficient matrices can be solved using the Thomas algorithm?
a) ⎡⎣⎢130246057⎤⎦⎥
b) ⎡⎣⎢100240357⎤⎦⎥
c) ⎡⎣⎢135046007⎤⎦⎥
d) ⎡⎣⎢135246357⎤⎦⎥