Computational Fluid Dynamics Questions and Answers for Aptitude test on “Explicit and Implicit Finite Difference Methods”.
1. Which of these methods of solving a system of equations will be needed after using an explicit scheme?
a) Sequential
b) Simultaneous
c) Iterative
d) Direct
Answer: a
Clarification: Explicit schemes result in marching solutions. Each step is dependent on the previous step only for one variable. The rest of the variables are found using the first obtained one. So, a simultaneous solution will not be needed here.
2. What is the main disadvantage of explicit schemes in a time-dependent problem?
a) Marching solution
b) Simultaneous equations
c) Small time-step size
d) Small grid size
Answer: c
Clarification: Explicit time-based schemes have a limited time-step size. Big time steps cannot be used. So, the total time of computation required to solve the system is very large when compared to the implicit schemes.
3. Implicit time-based problems will result in __________
a) Coupled equations
b) Uncoupled equations
c) Linear equations
d) Non-linear equations
Answer: a
Clarification: Implicit time-dependent solutions do not have a single unknown in a new time step. All the variables at a time step are coupled. So, they must be solved simultaneously to get the variables.
4. Which of these properties limit the time-step size in the explicit schemes?
a) Convergence
b) Stability
c) Consistency
d) Error
Answer: b
Clarification: The time step-size of an explicit scheme cannot be big. They are limited by the stability criterion. If the time-step size is bigger than the limit given by this criterion, the results will go extremely unstable.
5. What is advantageous in implicit schemes?
a) Error
b) Consistency
c) Convergence
d) Stability
Answer: d
Clarification: Implicit schemes do not have any restriction for the time-step size. They are stable for large time-steps also. Some of the implicit schemes are even unconditionally stable. Stability problems do not arise in implicit schemes.
6. Which of these is correct regarding implicit schemes?
a) Truncation error is less
b) Computation time is more
c) Time-step size is small
d) Easy to set-up
Answer: a
Clarification: As the time-step size is very large, the truncation error may become large and the accuracy of results may be less when compared to that of the explicit schemes. The total time of computation is less. But the algorithm is difficult to set-up.
7. Which of these may cause a problem to implicit schemes?
a) Coupled equations
b) Partial differential equations
c) Non-linear equations
d) Linear equations
Answer: c
Clarification: Though the implicit scheme has a great advantage of larger time steps, each step in an implicit scheme is large and takes more computational time. If the equation is non-linear, solving the system simultaneously will become more difficult. Usually, for these cases, the equations are linearized.
8. The time-step size in explicit schemes depends upon _____________
a) Grid size
b) Number of iterations
c) Total time interval
d) Given mathematical equation
Answer: a
Clarification: There is a limit posed to time-step size in explicit schemes. This limit depends on the grid size chosen. Once, the grid size is chosen, from the formula given by stability criterion, the maximum possible time-step size can be calculated.
9. Which of these schemes will lead to an implicit problem?
a) Higher-order schemes
b) SIMPLE algorithm
c) High-resolution scheme
d) Crank-Nicolson scheme
Answer: d
Clarification: Crank-Nicolson scheme is used to solve problems governed by parabolic equations. They result in implicit time-dependent problems. In CFD, they are usually used for finite difference solutions of boundary layer problems.
10. Consider the one-dimensional heat conduction equation. Apply forward difference method to approximate time rate and central difference method to approximate x-derivative. The resulting equation is in _____________
a) Implicit linear form
b) Explicit linear form
c) Explicit non-linear form
d) Implicit non-linear form
Answer: b
Clarification: The one-dimensional heat conduction equation is
(frac{partial T}{partial t}=α frac{partial^2 T}{partial t^2} )
Applying the difference approximations,
(frac{T_i^{n+1}-T_i^n}{Delta t}=alpha frac{T_{i+1}^n-2T_i^n+T_{i-1}^n}{(Delta x^2)} )
(T_i^{n+1}=T_i^n+alphafrac{Delta t(T_{i+1}^n-2T_i^n+T_{i-1}^n)}{(Delta x^2)} )
The equation is in explicit linear form.