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# Computational Fluid Dynamics Q – Discretization Aspects – Thomas Algorithm

1 - Question

Thomas algorithm is a ___________

a) Linear equations solver

b) Quadratic equations solver

c) Discretization method

d) Linear least square system

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Answer: aExplanation: 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

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Answer: dExplanation: 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

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Answer: cExplanation: 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

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Answer: bExplanation: 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

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Answer: cExplanation: 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

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Answer: cExplanation: 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

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Answer: bExplanation: 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

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Answer: cExplanation: 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

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Answer: dExplanation: 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⎤⎦⎥

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Answer: aExplanation: Thomas algorithm can be used to solve tri-diagonal matrices only. ⎡⎣⎢130246057⎤⎦⎥is a tri-diagonal matrix. ⎡⎣⎢100240357⎤⎦⎥ is an upper triangular matrix.⎡⎣⎢135046007⎤⎦⎥ is a lower triangular matrix. ⎡⎣⎢135246357⎤⎦⎥ is a square matrix.