This set of Computational Fluid Dynamics Multiple Choice Questions & Answers (MCQs) focuses on “Discretization Aspects – Errors and Stability Analysis”.

1. Which of these conditions is unstable?

a) Error is amplified in increasing iterations

b) Error is decreasing in increasing iterations

c) Error is amplified in decreasing iterations

d) Error is maintained in increasing iterations

View Answer

Explanation: A system is said to be unstable if the error increases with the increasing iterations. So, for a system to be stable, the error should decrease or at least should be maintained in further iterations.

2. The difference between the exact analytical solution of a partial differential equation and its numerical solution is as ___________

a) round-off error

b) discretization error

c) iteration error

d) modelling error

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Explanation: Discretization error is the error arising due to the numerical solution of a partial differential equation. It includes the error due to the numerical approximation of the governing equations and the boundary conditions.

3. The error due to the discretization of the partial differential equation is called as ______________

a) round-off error

b) discretization error

c) truncation error

d) iteration error

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Explanation: Truncation error arises when partial differential equations are approximated. Usually, the partial differential equations are approximated using a series expansion of infinite terms. The higher order terms are cut-off while approximating this series. So, it is called truncation error.

4. If the order of a discretized equation is ‘k’, what does it mean?

a) The last term of the equation is of (k+1)^{th} power

b) The last term of the equation is of k^{th} power

c) Truncation error is proportional to (k-1)^{th} power

d) Truncation error is proportional to k^{th} power

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Explanation: When a discretized equation is said to be of the order ‘k’, it means that the last term has (k-1)

^{th}power and the first term of the truncated equation has k

^{th}power. For example, in spatial derivative, the first term of the truncated equation will be proportional to (∆x)

^{k}.

5. ___________ become significant after a repeated number of calculations.

a) Round-off errors

b) Discretization errors

c) Truncation errors

d) Modelling errors

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Explanation: Round-off errors are introduced because of the round-off results produced by computers for a particular problem. When these round-off values are used for further calculations, they become significant after a certain time.

6. Round-off errors are important in ____________

a) modelling

b) iterations

c) discretization

d) truncation

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Explanation: Iterations are where the same process is repeated with the last generated value. This way, while iterating, rounding off the results become significant as they affect further iterations. Round-off errors are aggregated here.

7. Which of these iterative processes is unstable? (Note: \(\epsilon_n\) is the error in the n^{th} iteration).

a) \(\frac{\epsilon_{n+1}}{\epsilon_n} = 0.5\)

b) \(\frac{\epsilon_{n+1}}{\epsilon_n} = 0.75\)

c) \(\frac{\epsilon_{n+1}}{\epsilon_n} = 1.25\)

d) \(\frac{\epsilon_{n+1}}{\epsilon_n} = 1\)

View Answer

Explanation: For a system to be stable, the error should be decreasing or at least maintained.

Representing this mathematically,

ε

_{n+1}≤ ε

_{n}

\(\frac{\epsilon_{n+1}}{\epsilon_n} ≤ 1\)

For an unstable system,

\(\frac{\epsilon_{n+1}}{\epsilon_n} > 1\)

This happens in \(\frac{\epsilon_{n+1}}{\epsilon_n} = 1.25\). So, this system is unstable.

8. In Von Neumann analysis, the solution is expanded using ____________

a) Laurent series

b) McLaurin series

c) Taylor series

d) Fourier series

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Explanation: Von Neumann stability analysis is otherwise called as Fourier stability analysis. It is a technique used to analyse the stability of linear partial differential equations. It is named as Fourier stability analysis as it is based on Fourier decomposition of numerical error.

9. The error occurring while approximating the physical problem is called as ____________

a) Modelling error

b) Physical error

c) Mathematical order

d) Iteration error

View Answer

Explanation: The difference between the physical flow and the exact solution of the mathematical model is termed modelling error. This occurs because of approximating the physical model. It is not possible to model the exact physical scenario happening.

10. Tolerance for iteration errors is usually based on _______________

a) convergence

b) residuals

c) stability

d) round-off error

View Answer

Explanation: A tolerance should be set for the convergence of iterations to say when to stop the iterations. To set this, tolerance residuals are analysed. Residual directly quantifies the error in the solution of a system.

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