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Right choice is (a) Normalized Weighing Factor

Explanation: The abbreviation NWF STANDS for Normalized Weighing Factor. The abbreviation DWF stands for Downwind Weighing Factor. These TWO are implicit methods of FORMULATING high-resolution schemes.

The CORRECT answer is (c) the total VARIATION does not increase with increasing time

For EXPLANATION I would SAY: Any numerical method is said to be total variation diminishing if the total variation diminishes with time. This means that the value of total variation should not increase with time. It can either decrease or remain the same.

Correct choice is (C) DWFf > 0.5

The EXPLANATION is: For values of DWFf LARGER than 0.5, results in a system with NEGATIVE diagonal coefficients. So, the system becomes unsolvable by iterative methods. This happens whenever Φf > 0.5(Φc+Φd).

Correct choice is (C) 0≤DWF≤1

Explanation: By USING DWF, the high-resolution estimate of \(\tilde{\phi_f}\, or\, \phi_f\) is REDISTRIBUTED between the upwind and the downwind nodes. As the value of Φf computed using Φc and Φc. The value of DWF always lies between 0 and 1.

Correct choice is (a) \(\frac{\tilde{\phi_f}-\tilde{\phi_c}}{1-\tilde{\phi_c}}\)

BEST EXPLANATION: The Downwind WEIGHING Factor is given by

DWFF=\(\frac{\phi_f-\phi_c}{\phi_d-\phi_c}\)

Normalizing this, we get

DWFf=\(\frac{\tilde{\phi_f}-\tilde{\phi_c}}{\tilde{\phi_d}-\tilde{\phi_c}}\)

But, the value of \(\tilde{\phi_d}\) is 1. So,

DWFf=\(\frac{\tilde{\phi_f}-\tilde{\phi_c}}{1-\tilde{\phi_c}}\)

Correct answer is (c) The NWF METHOD is more robust than the DWF method

The EXPLANATION is: The NWF method is created to overcome the shortcomings of the DWF method. In the NWF method, instabilities do not arise. This ALLOWS the NWF method to be much more robust than the DWF method.

The correct answer is (B) 1

The explanation: The RELATION between the DWF formulation and the TVD formulation is given by

DWFf=\(\FRAC{1}{2}\) ψ(rf)

The ψ(rf) value for downwind scheme is 2. Therefore, the DWFf value is 1.

Correct answer is (b) NWF and DWF

For explanation I would SAY: The convergence problem of the high-resolution schemes paved the WAY for techniques of IMPLEMENTING high-resolution schemes that are more implicit such as NWF and DWF.

Correct answer is (a) (Ψ,r)

EXPLANATION: A Sweby’s diagram is used to REPRESENT the TVD. This diagram is DRAWN with the flux LIMITER (Ψr) in the y-direction and the variable r in the x-direction. A high-resolution scheme should lie in a particular REGION of this diagram to be TVD and monotonic.

Correct option is (d) TV^t=∑iΦ(i+1)Φi

Explanation: Total VARIATION is the SUMMATION of variations of the flow variable between two consecutive NODES. This is MATHEMATICALLY given as

TV^t=∑iΦi+1-Φi.

The correct answer is (c) undershoots and overshoots

Easiest EXPLANATION: The major shortcoming of the higher order schemes is the UNBOUNDEDNESS. This can be seen in the QUICK scheme. They tend to produce undershoots or overshoots and even oscillations NEAR the sudden jumps or STEEP gradients in the VARIABLE.

Right answer is (a) UPWIND BIASED

The BEST explanation: Physically, CONVECTION TRANSPORTS fluid from upstream to downstream. So, the numerical convection schemes should also be upwind biased. This is why, the NVF approach involved far upwind node also but not the far downwind node.

Correct option is (a) a HIGHER order bounded SCHEME

Explanation: The high-resolution scheme is a COMBINATION of a higher-order profile and the CONVECTION Boundedness Criterion (CBC) to get the most out of it. So, it can be called a higher order bounded scheme.

The correct OPTION is (d) continuous

Easy explanation: For a scheme to be bounded, the relationship between the KNOWN and the unknown variable should be continuous. REPRESENTING in the NVF TERMS, the function \(\tilde{\phi_f}=F(\tilde{\phi_c})\) of a scheme should be continuous.

Correct choice is (c) Sweby’s diagram and NVD

The EXPLANATION: The NVD (Normalized Variable Diagram) and Sweby’s diagram are USED for visualizing the high-resolution schemes in the NVF and TVD formulation respectively. They can be used to visualize the flux limiters.

Right answer is (b) first-order accurate

The best I can EXPLAIN: GODUNOV theorem proved that any monotonic linear scheme cannot be higher-order. It will always be first order. THEREFORE, all the higher-order linear SCHEMES will be non-monotonic. This is why limiters are used to construct high resolution schemes.

Correct CHOICE is (d) points (0,0) and (1,1)

Explanation: In the NVD diagram, the MONOTONIC PROFILE in the range 0\(\leq\widetilde{\phi_c}\leq\)1 should CONNECT the points (0,0) and (1,1) for a high-resolution scheme to be CONSTRUCTED. For the other ranges, the profile should follow the upwind scheme.

Right option is (d) TVD method

The BEST explanation: The deferred correction, Downwind WEIGHING Factor (DWF) and UPWIND Weighing Factor (UWF) are all METHODS to implement the higher-order and high-resolution schemes. TVD scheme is not used for this purpose.

Right answer is (a) 0 and 2

To elaborate: The TVD schemes are developed starting from the upwind scheme. If flux limiters are used, the upwind scheme will CHANGE its nature. So, no flux limiter is required in this case and hence flux limiter for an upwind scheme is 0. For DOWNWIND scheme, the whole PROFILE in the Sweby’s DIAGRAM should be in the line Ψ=2. So, the flux limiter here is 2.

The correct option is (b) the ratio of two consecutive gradients

Best EXPLANATION: Flux limiter prevents the excessive use of flux in regions where oscillations might OCCUR and maximizes the contribution in SMOOTH areas. The flux limiter is denoted by Ψ(R), where r is usually TAKEN as the ratio of two consecutive gradients.

Correct option is (a) 1 and 0

Easiest EXPLANATION: From the normalization formula,

\(\TILDE{\phi_d}=\frac{\phi_d-\phi_u}{\phi_d-\phi_u}=1\).

\(\tilde{\phi_u}=\frac{\phi_u-\phi_u}{\phi_d-\phi_u}=0\).

The correct answer is (b) avoid sharp angles

Explanation: A high-resolution scheme can have COMPOSITE nature of profiles in the NVD frame which involves CUT. But, at these cuts, the angle should not be sharp for the SCHEMES to have better CONVERGENCE.

Right answer is (b) Problems involving shocks

To explain: A normal first-order scheme produces ACCURATE results for flows involving high discontinuities like shocks at the cost of INCREASED GRID points. This PROBLEM is overcome by high-resolution schemes.

The CORRECT option is (c) the UPWIND scheme

To EXPLAIN: Deferred correction procedure is used to overcome PROBLEMS related to the convergence of a scheme. The coefficients used here are based on the upwind scheme which is suitable for CONVECTION.

Right option is (b) (l,K)

To EXPLAIN: ALONG the downwind line of NVD, the values of (l,k)=(0,1), a value of zero is obtained for the diagonal coefficient and the system BECOMES unstable. To overcome this problem, the values of (l,k) are set equal to (L,1-LΦf). The value of L can be chosen which is usually set to l in the previous INTERVAL.

Right OPTION is (a) the flux limiter

To explain: A TVD scheme should not be completely upwind or downwind. So, to DEVELOP a TVD scheme, an approach is USED in which a PORTION of the anti-diffusive flux is added to the upwind scheme. This flux is limited by a flux limiter function. To find the best flux limiter is the work in developing a TVD scheme.

Right option is (d) \(\FRAC{1}{2}\)

To EXPLAIN: For the CENTRAL difference scheme,

ψ(rf)=1

So, the VALUE of DWFf for this scheme is ½.

Correct CHOICE is (a) Difference between the high-resolution and the UPWIND schemes

The explanation is: The source term of the given flow problem is CONSIDERED. In addition to this, a source term is added in the deferred correction method which is EQUAL to the difference between the high-resolution scheme and the upwind scheme.

Right ANSWER is (a) FAR downwind node

Easiest explanation: The NVF is a FACE formulation procedure BASED on normalizing the dependent variable for which the flow variable at the face has to be CONSTRUCTED. The NVF approach relies upon the upwind, downwind and far upwind nodes.

Correct choice is (c) QUICK scheme

Explanation: The SMART scheme is CONSTRUCTED USING the QUICK scheme. The SMART scheme can be easily modified to GET a higher convergence. This MODIFICATION is done by modifying the vertical portion of the PROFILE of the SMART scheme.

Correct CHOICE is (a) Convergence and stability

For explanation: The difficulty ARISING when explicitly expressing a FLOW variable in terms of NEIGHBOURING values is the convergence RATE. They violate one of the basic rules for stability and convergence which is the coefficients should be of the same sign.

Correct option is (c) is low

To explain I would say: If the difference between the high-resolution scheme line and the upwind line INCREASES, convergence is AFFECTED. The rate of convergence DIMINISHES. This is when the normalized difference between the cell FACE values is high.

The correct answer is (c) Ψr=min⁡(2R,r) & if r>0; Ψr=0 & if r≤0

For explanation I would say: Similar to the Convection BOUNDEDNESS CRITERION, a flux limiter should satisfy the following criterion to be a TVD. There is a list of conditions which has to be satisfied. Simplifying and combining all of them, we get

\(\Psi = \left\{\begin{matrix}

min(2r,r) & r>0 \\

0 & r\leq 0

\end{matrix}\right\}.\)

The correct answer is (b) the normalized DIFFERENCE between the cell face VALUES

The explanation: The effects of the DEFERRED correction method can be SEEN in an NVD FRAME. The normalized difference between the cell face values is given by the difference between the upwind and the high-resolution scheme lines in the NVD.

The correct answer is (b) NVF

To explain I would SAY: The NWF method operates by linearizing the normalized interpolation profile. This uses the simplified FORM of the TVD FRAMEWORK given by \(\tilde{\PHI}_f=a\tilde{\phi}_c+b\). The correction term here is smaller than the one obtained by standard DEFERRED correction methods.