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# Computational Fluid Dynamics – Transient Flows – Predictor-Corrector and Multipoint Methods

1 - Question

The predictor-corrector method is a combination of ______________

a) midpoint and trapezoidal rules

b) backward Euler method and Trapezoidal rule

c) implicit and explicit methods

d) forward and backward Euler methods

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Answer: cExplanation: Explicit methods are very easy to program and they need less computational cost. But they are not stable. The implicit methods are unconditionally stable but more expensive and iterative. So, the positives of both of these methods are combined by the predictor-corrector method.

2 - Question

In the two-level predictor-corrector method, the prediction is done using _____________

a) trapezoidal rule

b) explicit Euler method

c) midpoint rule

d) implicit Euler method

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Answer: bExplanation: The prediction step uses the forward (explicit) Euler method. The formula for this method is given as Φn+1*=Φn+f(tn,Φn )Δt. Here, ^*indicates that this answer is not the final one. This predicted value is corrected later

3 - Question

Which of these formulae is used for the corrector step of the two-level predictor-corrector method?

a) Φn+1=Φn+13 [2f(tn,Φn )+f(tn+1,Φn+1*)]Δt

b) Φn+1=Φn+12 [2f(tn,Φn )+f(tn+1,Φn+1*)]Δt

c) Φn+1=Φn+13 [f(tn,Φn )+2f(tn+1,Φn+1*)]Δt

d) Φn+1=Φn+12 [f(tn,Φn )+f(tn+1,Φn+1*)]Δt

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Answer: dExplanation: The predicted result of the two-level prediction-correction method is corrected using the trapezoidal rule. The trapezoidal rule uses the linear interpolation between two time points. The formula is Φn+1=Φn+12 [f(tn,Φn )+f(tn+1,Φn+1*)]Δt.

4 - Question

The two-level predictor-corrector method is __________

a) second-order accurate

b) first-order accurate

c) fourth-order accurate

d) third-order accurate

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Answer: aExplanation: The order of accuracy of the two-level predictor-corrector method is the same as the trapezoidal rule. This is because they employ the trapezoidal rule. The trapezoidal rule is second-order accurate.

5 - Question

The stability of the two-level predictor-corrector method matches with that of the __________

a) midpoint rule

b) trapezoidal rule

c) backward Euler method

d) forward Euler method

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Answer: dExplanation: The predictor-corrector method takes the stability of the explicit Euler method. Though this is not advantageous, at least, the accuracy is better for the two-level predictor-corrector method.

6 - Question

The predictor-corrector method is maximum ___________

a) second-order accurate

b) cannot be defined

c) third-order accurate

d) fourth-order accurate

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Answer: bExplanation: The two-level predictor-corrector method has the highest possible order of accuracy as two. But, in general, there are many predictor-corrector methods are of a higher-order. So, the order of accuracy cannot be decided before.

7 - Question

To increase the order of accuracy, the multipoint method uses ___________

a) highly stable two-level methods for prediction and correction

b) higher-order two-level methods for prediction and correction

c) additional points where data is already available

d) additional points where data is interpolated

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Answer: cExplanation: To increase the order of accuracy of the predictor-corrector method, multiple points are used instead of two points. The extra points are those obtained from the previous calculations.

8 - Question

The Adam-Bashforth method is ____________

a) an explicit method

b) an implicit method

c) a first-order accurate method

d) a second-order accurate method

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Answer: aExplanation: The Adam-Bashforth method is a multipoint method. Therefore, its order of accuracy will be more than two. It is an explicit method of approximation. But it is a lot more advantageous than the explicit Euler method.

9 - Question

Which of these is used by the Adam-Bashforth method?

a) Newton’s method

b) Frobenious covariant

c) Frobenious norm

d) Lagrange polynomial

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Answer: dExplanation: A Lagrange polynomial is used to get the polynomials at the required number of points in the Adam-Bashforth method. The order of accuracy of this method depends on the number of polynomials used.

10 - Question

Which of these is correct for the multipoint method?

a) multiple derivatives at each time step

b) only one evaluation of derivative per time step

c) order of accuracy is restricted to four

d) extremely unstable

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Answer: bExplanation: The multipoint approach has a lot of advantages. The main advantage is that any order of accuracy can be obtained by using a different number of points. Also, at one time-step, only one new derivative has to be evaluated. Others are those stored from the older time-steps.