This set of Computational Fluid Dynamics Multiple Choice Questions & Answers (MCQs) focuses on “Transient Flows – Two-level Methods”.

1. For which kind of problems are the two-level methods used?

a) Spatial integrations

b) Spatial problems in ODEs

c) Temporal initial value problems in ODEs

d) Temporal initial value problems in integration

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Explanation: The two-level methods are used to discretize the ordinary differential equations which of the initial value types. They proceed using two steps in time starting from the available initial value.

2. Which of these methods will not come under a two-level method?

a) Forward Euler method

b) Adams method

c) Trapezoidal method

d) Midpoint rule

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Explanation: The Adams method is a multipoint method. Some of the two-level methods are the forward and backward Euler methods, the midpoint rule and the trapezoidal rule. They do not use more than two points to solve the system.

3. Which of these methods is the basis of the leapfrog method?

a) Midpoint rule

b) Trapezoidal rule

c) Implicit Euler method

d) Explicit Euler method

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Explanation: The midpoint rule uses the midpoint of the interval to approximate the results. This forms the basis of the leapfrog method which is a very important method for solving the partial differential equations.

4. Which of these methods is derived from the trapezoidal rule?

a) Euler method

b) Adams method

c) Runge-Kutta method

d) Crank-Nicolson method

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Explanation: The Crank-Nicolson method is another method for solving the partial differential equations. They are derived from the trapezoidal rule of numerical approximation. The trapezoidal rule is a straight point interpolation.

5. Which of these is an explicit method of solving initial value problems?

a) Forward Euler method

b) Adams method

c) Trapezoidal method

d) Midpoint rule

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Explanation: The forward Euler method needs the value of the flow variable at the endpoint. Therefore, it cannot be calculated without any interpolation or approximation. This is an explicit method.

6. What is the condition of stability for the forward Euler method when the function is real?

a) \(\frac{\Delta t\partial f}{\partial\phi} <2\)

b) \(\big|\frac{\Delta t\partial f}{\partial\phi}\big|<2\)

c) Always stable

d) Never statble

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Explanation: The forward Euler method is conditionally stable. For this method to be stable, it needs the following condition to be satisfied.

\(\big|1+\frac{\Delta t\partial f}{\partial\phi} \big|<1\)

When the function f is real, this becomes

\(\big|\frac{\Delta t\partial f}{\partial\phi} \big|<2\).

7. The trapezoidal rule is ___________

a) stable when Δ t>1

b) stable when Δ t<1

c) always stable

d) never stable

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Explanation: The trapezoidal rule, the midpoint rule and even the backward implicit Euler method are all unconditionally stable. They do not need any condition for the solution to be bounded when the input is bounded.

8. Which of these methods is stable for non-linear systems?

a) Forward Euler method

b) Backward Euler method

c) Trapezoidal method

d) Midpoint rule

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Explanation: Though the trapezoidal rule is unconditionally stable, it is not the case for non-linear problems. But, the backward Euler method behaves well and smooth for non-linear systems also. They produce smooth results for large time steps too.

9. What is the order of accuracy of the forward Euler method?

a) First-order

b) Second-order

c) Third-order

d) Fourth-order

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Explanation: The forward Euler method has the least accuracy in the two-level methods of approximating ODEs. They are first-order accurate. But, the Taylor series expansion of the forward Euler method says it to be a second-order accurate scheme.

10. What is the maximum possible accuracy for the two-level methods?

a) Fifth-order

b) Fourth-order

c) Third-order

d) Second-order

View Answer

Explanation: The two-level schemes can at most give an accuracy of order two. The trapezoidal and midpoint rules and also the backward Euler method are second-order accurate. But, this does not determine the accuracy of the method solely.

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