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# Computational Fluid Dynamics – CFD Techniques – Alternating Direction Implicit Techniques

1 - Question

Which of these is important while solving a system with implicit methods?

a) Linearizing the difference equation

b) Linearizing the partial differential equation

c) Normalizing the difference equation

d) Normalizing the partial differential equation

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Answer: aExplanation: The nature of the original PDE should not be changed while solving the problem. If the problem is non-linear in PDE, to make it solvable by the implicit scheme, the difference equation should be linearized.

2 - Question

How many steps does the Alternating Direction Implicit (ADI) scheme involve?

a) One step

b) Two steps

c) Three steps

d) Four steps

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Answer: bExplanation: The whole process is to initiate from the known values at the time-step t and move on to the required time-step t+Δ t. For this marching, the ADI scheme uses two steps. The solution does not directly move to the final step t+Δ t.

3 - Question

The intermediate step of the ADI scheme is at __________

a) t+Δt3

b) t+Δt2

c) t-Δt2

d) t+Δt4

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Answer: bExplanation: The first step of the ADI scheme is from t to t+Δt2. The second step of the ADI scheme is from t+Δt2 to t+Δt. Here, the intermediate step t+Δt2 is extra for the process and the results here are not actually needed.

4 - Question

Which of these statements is correct about the first step of the ADI scheme?

a) x-derivative is treated implicitly

b) y-derivative is treated implicitly

c) Time derivative is treated implicitly

d) Thomas algorithm is not used

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Answer: aExplanation: In the first step with the time interval Δt2, the spatial derivatives are replaced using the central difference scheme. Only the x-derivative is treated implicitly. Then the resulting equations are solved using the TDMA method.

5 - Question

The second step of the ADI scheme is swept over __________ direction.

a) both the x and y

b) the x

c) the y

d) the time

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Answer: cExplanation: The second step of the ADI scheme uses the time domain from t+Δt2 to t+Δt. Here, the y-derivative is treated implicitly after replacing the derivatives with central differences. The solutions are swept in the y-direction here.

6 - Question

If there are N grid points in both the x and y-directions, how many times does the ADI scheme use the Thomas algorithm?

a) N/2

b) 2

c) N2

d) 2N

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Answer: dExplanation: In the first step of the ADI scheme, the Thomas algorithm is used N times to solve in the x-direction. Similarly, in the y-direction, again the Thomas algorithm is used N times to get the solutions. Totally, the Thomas algorithm is used 2N times.

7 - Question

The order of accuracy of the ADI scheme in the time direction is ___________

a) third-order

b) fourth-order

c) first-order

d) second-order

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Answer: dExplanation: The truncation error of the ADI scheme in the time direction is of order two. The ADI scheme is second-order accurate in the time direction. It uses two time-steps to move from the step t to t+Δt.

8 - Question

The order of accuracy of the ADI scheme in the x and y-directions are __________ and __________

a) second-order and first-order

b) first-order and second-order

c) second-order and second-order

d) second-order and third-order

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Answer: cExplanation: The truncation errors of the ADI scheme in the x and the y-directions are O(Δx2) and O(Δy2). Therefore, the order of accuracy of the ADI scheme in the x and the y-directions are two and two respectively.

9 - Question

The ADI scheme is particularly suitable for ____________ problems.

a) parabolic and elliptic

b) parabolic and hyperbolic

c) hyperbolic

d) parabolic

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Answer: aExplanation: The ADI scheme is useful to solve many fluid flow problems including the heat conduction and the mass diffusion problems. It is particularly suitable for the parabolic and elliptic problems.

10 - Question

Which of these is a popular version of the ADI scheme?

a) Operator splitting

b) Approximate factorization

c) ALU algorithm

d) SIMPLE algorithm

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Answer: bExplanation: The major objective of the ADI scheme is to make the Thomas algorithm for tri-diagonal matrices applicable to multi-dimensional problems. This scheme has other versions too. One of the popular versions of the ADI scheme is the approximate factorization scheme