This set of Computational Fluid Dynamics Multiple Choice Questions & Answers (MCQs) focuses on “Errors in Finite Difference Approximations”.

1. Which is the major error occurring due to the finite difference approximations?

a) Discretization error

b) Round-off error

c) Iteration error

d) Modelling errors

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Explanation: The major error occurring in the finite difference method is the discretization error. This error occurs due to both temporal and spatial discretization using an approximation for the discretization. This is also called a numerical error.

2. What is the source of discretization error in the finite difference method?

a) Numerical error

b) Round-off error

c) Truncation error

d) Modelling error

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Explanation: Discretization error occurs because of the truncation errors which arise while discretizing the PDEs. It is named truncation error as the root cause of it is the truncation of the higher order terms in the series expansion.

3. The exact solution of the partial differential equation varies from the exact solution of the discretized equations by ___________

a) truncation error

b) discretization error

c) iteration error

d) modelling error

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Explanation: The difference between the exact solution of the partial differential equation and the solution of the algebraic equation is the discretization error. Mathematically

Φ=Φ

_{num}+∈

_{d}

Where,

Φ → Exact solution

Φ

_{num}→ Numerical solution

∈

_{d}→ Discretization error

4. Truncation error is the difference between __________

a) the exact solution of the partial differential equation and the discretized equations

b) the exact partial differential equation and the discretized equations

c) the exact solution and the numerical solution of the partial differential equations

d) the exact partial differential equation and its solution

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Explanation: Truncation error is the difference between the exact partial differential equation and the discretized algebraic equation. This arises as we cut-off the higher order terms in the Taylor series expansion.

5. Information about the magnitude and distribution of the truncation error can be useful for ____________

a) correcting the error

b) increasing the stability

c) refining the grid

d) converging the solution

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Explanation: It is not possible to decrease the discretization error as it will lead to increased algebra and computation. So, the information about the discretization error can be used only for refining the girds and achieve the same level of discretization everywhere in the solution.

6. How is the discretization error found?

a) Difference between the solutions obtained from systematically refined grids

b) Difference between the exact and the numerical solutions

c) Difference between the exact solution and the solution from the refined grid

d) Difference between the coarse grid solution and the exact solution

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Explanation: Discretization error is originally the difference between the exact solution of the partial differential equation and the solution of the algebraic equation. But, as the exact solution is not known, we estimate the discretization error as the difference between the solutions obtained from systematically refined grids.

7. What is Richardson extrapolation used for?

a) To increase the accuracy

b) To decrease the error

c) To create convergence monotony

d) To increase the rate of convergence

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Explanation: Richardson extrapolation is a sequence acceleration method. It is used to increase the rate of convergence of a system. In the finite difference method, it is used to find accurate results from the discretized results.

8. What does Richardson extrapolation do in finite difference schemes?

a) Add the error estimate to the results of the finest grid

b) Subtract the error estimate from the results of the finest grid

c) Add the error estimate to the results of the current grid

d) Subtract the error estimate from the results of the current grid

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Explanation: When we have the results of many grid arrangements ranging from coarse to fine. A solution which is more accurate than the solution of the finest grid can be obtained by adding the error estimate to the results of the finest grid available.

9. When is the Richardson extrapolation accurate?

a) When the system is stable

b) When the convergence is monotonic

c) When the system is consistent

d) When the system is linear

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Explanation: The Richardson extrapolation is very useful as it is a simple method. But, it can give accurate results only when the convergence is monotonic. This convergence represents the convergence of error while refining the grid.

10. What happens when the convergence is not monotonic?

a) Solutions will always converge

b) Solutions will not converge

c) Erroneous solutions may converge

d) Error will increase

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Explanation: The order of convergence is valid only when the convergence is monotonic. This is because, for two consecutive grids, results may also converge even if the error is large. Then, a third grid arrangement must be used to assure the real convergence of the solution.

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