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# Computational Fluid Dynamics – Discretization Aspects – Stability

1 - Question

Stability is defined _________

a) only for iterative solvers

b) only for direct solvers

c) for all numerical solvers

d) for all discretization processes

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Answer: aExplanation: Only for iterative solvers, stability can be defined. It describes how well the equations can be solved using iterative solvers. It needs the solver not to diverge the solution from the exact answers.

2 - Question

Which of these is used to analyse the stability of a system?

a) Nusselt number

b) Courant number

c) Peclet number

d) Von Neumann’s method

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Answer: dExplanation: Von Neumann’s method is a widely used method of analysing the stability of any mathematical system. Since CFD uses numerical methods to solve problems, Von Neumann’s method is applicable to the CFD schemes also.

3 - Question

Stability is the property of a _________

a) partial differential equation

b) discretized equation

c) discretization process

d) mathematical model

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Answer: bExplanation: Stability is not the property of a discretization process; it is the property of the resulting system of discretized equation. This can be analysed mathematically for the system. Stability does not exist for a partial differential equation or a mathematical model.

4 - Question

For which of these problems, the error will be bounded if the system is stable?

a) Transient problems

b) Subsonic problems

c) Supersonic problems

d) Inviscid problems

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Answer: aExplanation: For transient problems, a stable system keeps the error bounded when time increases. Stability for a transient problem has special characteristics. Here, stability guarantees that the method gives a bounded solution if the exact solution is bounded.

5 - Question

A system is said to be stable if _________

a) the results for different boundary and initial conditions are different

b) the results for different boundary and initial conditions are the same

c) the system can be solved for different initial and boundary conditions

d) the result of two consequent iterations are the same

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Answer: cExplanation: A stable system of algebraic equation means that the system can be solved for different boundary conditions and initial conditions to get the flow properties. While varying the boundary conditions, the system should not become unsolvable.

6 - Question

Stability of explicit transient schemes is related to _________

a) over-relaxation

b) the time-step

c) the grid size

d) under-relaxation

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Answer: bExplanation: Stability of transient schemes vary for implicit and explicit problems. Stability of explicit transient scheme is ensured by limiting the time-step size. Stability of implicit transient scheme is improved by under-relaxing the equations.

7 - Question

Which of the following is a sufficient condition for a system to be stable?

a) Gauss criterion

b) Convergence criterion

c) Stopping criterion

d) Scarborough criterion

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Answer: dExplanation: A system of linear equations can be taken as stable if it satisfies the Scarborough criterion. Scarborough criterion gives a condition about the coefficient matrix of the algebraic system. This criterion also gives information about the boundedness of the problem.

8 - Question

What is the Scarborough criterion?

a) The coefficient matrix has larger values in the main diagonal

b) The coefficient matrix has larger values above the main diagonal

c) The coefficient matrix has larger values below the main diagonal

d) The coefficient matrix has larger values except the main diagonal

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Answer: aExplanation: Scarborough criterion needs the coefficient matrix to be diagonally dominant. This means that the diagonal elements should be larger than the non-diagonal elements for a particular row in the coefficient matrix.

9 - Question

A system is said to be stable if _________

a) it does not magnify the errors occurring in the course of solution

b) it does not converge

c) it reduces the errors occurring in the course of solution

d) it does not maintain the errors occurring in the course of solution

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Answer: aExplanation: A mathematical system is said to be stable if the system does not go on increasing the errors produced while solving the problem. So, the solution should not diverge from the exact answer though errors are present.

10 - Question

It is difficult to analyse the stability of _________

a) non-linear systems without boundary conditions

b) linear systems with boundary conditions

c) non-linear systems with boundary conditions

d) linear systems without boundary conditions

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Answer: cExplanation: While solving non-linear coupled equations with boundary conditions, analysing stability is difficult. So, the stability of a system is usually analysed for linear problems without boundary conditions. For non-linear systems, we rely on experience to know its stability.