1. 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
Explanation: 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. How many steps does the Alternating Direction Implicit (ADI) scheme involve?
a) One step
b) Two steps
c) Three steps
d) Four steps
Explanation: 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. The intermediate step of the ADI scheme is at __________
a) t+\(\frac{\Delta t}{3}\)
b) t+\(\frac{\Delta t}{2}\)
c) t-\(\frac{\Delta t}{2}\)
d) t+\(\frac{\Delta t}{4}\)
Explanation: The first step of the ADI scheme is from t to t+\(\frac{\Delta t}{2}\). The second step of the ADI scheme is from t+\(\frac{\Delta t}{2}\) to t+Δt. Here, the intermediate step t+\(\frac{\Delta t}{2}\) is extra for the process and the results here are not actually needed.
4. 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
Explanation: In the first step with the time interval \(\frac{\Delta t}{2}\), 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. 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
Explanation: The second step of the ADI scheme uses the time domain from t+\(\frac{\Delta t}{2}\) 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. 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
Explanation: 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. 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
Explanation: 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. 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
Explanation: 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. The ADI scheme is particularly suitable for ____________ problems.
a) parabolic and elliptic
b) parabolic and hyperbolic
c) hyperbolic
d) parabolic
Explanation: 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. Which of these is a popular version of the ADI scheme?
a) Operator splitting
b) Approximate factorization
c) ALU algorithm
d) SIMPLE algorithm
Explanation: 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.