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# Computational Fluid Dynamics – Discretization Aspects – Convergence

1 - Question

Convergence is defined for _________

a) Elimination method

b) Iterative solvers

c) Direct solvers

d) Cramer’s method

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Answer: aExplanation: Convergence is a property of the iterative solvers used for solving the discretized system of equations. It cannot be defined for direct solvers as they do not have repeated steps and similar answers.

2 - Question

A solution is ideally converged if _________

a) the results match with the exact solution

b) the results for two consecutive iterations are the same

c) the results for two schemes are the same

d) the results for different boundary conditions are the same

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Answer: bExplanation: Ideally, a solution of a system of equations is said to be converged if the results of two consecutive iterations are exactly the same without any variation. No more iterations are required after this.

3 - Question

In real, how is convergence defined?

a) Variations are accepted

b) When the variation is less than the result

c) When the variation falls below a certain acceptable range

d) When the variation is the same as the result

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Answer: cExplanation: In real, the variation between two consecutive iterations cannot be exactly the same. The value of variation will be constantly decreasing. So, the solution is said to be converged when the range of variation is acceptable.

4 - Question

Convergence decides _________

a) the result of the numerical method

b) the method of iteration

c) the stability of the system

d) when to stop the iterations

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Answer: dExplanation: Iterative processes start with an initial guessed answer. This converges into the correct result as the number of iterations increases. Convergence criterion says when to stop this repeated process with acceptable error.

5 - Question

How is the tolerance of convergence decided?

a) Based on stability and consistency

b) Based on efficiency and accuracy

c) Based on efficiency and consistency

d) Based on consistency and accuracy

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Answer: bExplanation: In practical, the results of two iterations does not exactly match with each other. The iterations are stopped when the solution reaches some acceptable tolerance. This tolerance is decided in a way that it affects neither the accuracy of the solution nor its efficiency.

6 - Question

If the tolerance value to stop the iteration is too big, which of these properties will be affected?

a) Accuracy

b) Efficiency

c) Stability

d) Conservativeness

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Answer: bExplanation: The tolerance should be a balance between both accuracy and efficiency. If the tolerance is too big, iterations will stop soon but the answers will not be accurate. On the other hand, if the tolerance is too small, the number of iterations will be more. This will make the solution inefficient.

7 - Question

Which of these statements is wrong?

a) Convergence is applicable for iteration processes

b) Convergence is affected by accuracy and efficiency

c) Converged solutions do not vary much with further iterations

d) Converged solutions are exact

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Answer: dExplanation: Converged solutions are just correct in respect to the iteration. We cannot say that all the solutions which converge are correct. The converged solutions may be wrong I respect to other properties. Convergence does not ensure correct solutions.

8 - Question

Which of these is related to convergence?

a) Stopping criteria

b) Peclet number

c) Lax Equivalence Theorem

d) Scarborough criteria

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Answer: cExplanation: Lax Equivalence Theorem gives the condition for stability. It is applicable only for the finite difference methods applied to linear initial value problems. This is not applicable to non-linear systems.

9 - Question

Which of these properties is not included in the Lax Equivalence Theorem?

a) Stability

b) Boundedness

c) Consistency

d) Convergence

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Answer: bExplanation: Lax Equivalence Theorem states that “For a well-posed linear initial value problem solved by the finite difference approximation which satisfies consistency condition, stability is the necessary and sufficient condition for convergence”.

10 - Question

For small grid sizes, convergence is related to _________

a) truncation error

b) stability

c) consistency

d) boundedness

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Answer: aExplanation: When the grid sizes are sufficiently small, convergence is related to truncation error. The rate of convergence is governed by the order of principal truncation error component which is used to approximate the partial differential equations.