250+ TOP MCQs on Discretization Aspects – Thomas Algorithm and Answers

Computational Fluid Dynamics Multiple Choice Questions on “Discretization Aspects – Thomas Algorithm”.

1. Thomas algorithm is a ___________
a) Linear equations solver
b) Quadratic equations solver
c) Discretization method
d) Linear least square system
Answer: a
Clarification: 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
Answer: d
Clarification: 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
Answer: c
Clarification: 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 i^th equation is a_i Φ_i=b_i Φ_(i+1)+c_i Φ_(i+1)+d_i. While solving this system using Thomas algorithm, we get Φ_i=P_i Φ_(i+1)+Q_i. What are P_i and Q_i?
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}})
Answer: b
Clarification: As given,
Φ_i = P_iΦ_i+1+Q_i
Φ_i-1 = P_i-1Φ_i+Q_i-1
The i^th equation is,
a_iΦ_i = b_i Φ_i+1 + c_iΦ_i-1 + d_i
a_iΦ_i = b_i Φ_i+1 + c_i(P_i-1Φ_i + Q_i-1) + d_i
a_iΦ_i – c_iP_i-1Φ_i = b_i Φ_i+1+c_iQ_i-1+d_i
Φ_i(a_i-c_i P_i-1) = b_iΦ_i+1+c_i Q_i-1+d_i
(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 i^th equation of a system of n equations be a_iΦ_i=b_i Φ_i+1+c_iΦ_i-1+d_i. Which of these is correct?
a) c_N=0; b_N=0
b) c_N=0; b₁=0
c) c₁=0; b_N=0
d) c₁=0; b₁=0
Answer: c
Clarification: Node 1 will not have a previous node (c₁=0). The last node will not have the next node (b_N=0).

6. Using the Thomas algorithm, if the i^th unknown is Φ_i=P_i Φ_i+1+Q_i. what is the last unknown value Φ_N equal to?
a) 0
b) P_N
c) Q_N
d) 1
Answer: c
Clarification: For the last equation, b_N=0. So, P_N=0. Φ_N=P_NΦ_N+1+Q_N=Q_N.

7. While solving a system of equations with the Thomas algorithm, in which order are the values of P_i and Q_i found?
a) Backwards
b) Forward
c) Simultaneously
d) Depends on the problem
Answer: b
Clarification: To find the current values of P_i and Q_i, the previous values P_i-1 and Q_i-1 should be known. So, this is started from the first equation and done in forward order.

8. After finding all the values of P_i and Q_i, in which order are the values of Φ_i found?
a) Forward
b) Simultaneously
c) Backwards
d) Depends on the problem
Answer: c
Clarification: The last value of Φ_N can be found using Q_N. Using this, the previous value is found using the formula Φ_i=P_iΦ_i+1+Q_i. So, it is done backwards.

9. Consider a system of equations where the i^th equation is a_iΦ_i=b_i Φ_i+1+c_iΦ_i-1+d_i. While solving this system using Thomas algorithm, we get Φ_i=P_iΦ_i+1+Q_i. 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})