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The correct answer is (a) Taylor Series

For explanation: Lax-Wendroff TECHNIQUE uses the Taylor series expansion to APPROXIMATE its TIME DERIVATIVES. This makes the technique marching in time in an explicit way. The NUMBER of terms used for this expansion decides the accuracy of this system.

Correct answer is (a) Sequential

For explanation: EXPLICIT schemes result in marching solutions. Each step is DEPENDENT on the previous step only for one variable. The rest of the variables are found using the FIRST obtained one. So, a SIMULTANEOUS solution will not be NEEDED here.

The CORRECT choice is (a) \(-(\frac{\partial^2 U}{\partial X^2})_{i,j}\frac{\DELTA x}{2}\)

EXPLANATION: The Taylor series expansion of ui+1,j is

\(u_{i+1,j}=u_{i,j}+(\frac{\partial u}{\partial x})_{i,j} \Delta x+(\frac{\partial ^2 u}{\partial x^2})_{i,j}\frac{(\Delta x)^2}{2}+⋯\)

\(\frac{u_{i+1,j}-u_{i,j}}{\Delta x}=(\frac{\partial u}{\partial x})_{i,j}+(\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{2}+⋯\)

\((\frac{\partial u}{\partial x})_{i,j}=\frac{u_{i+1,j}-u_{i,j}}{\Delta x}-(\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{2}-…\)

The term –\((\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{2}-… \)is truncated. So, the first term of truncation error is –\((\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{2}\).

Correct option is (a) Rearward differences

Easy explanation: If forward differences are USED in the predictor step, rearward differences should be used in the corrector step and VICE versa. At every time-step, this SEQUENCE should be changed while SOLVING a time-marching problem.

Right answer is (B) Stability

The best I can explain: The time step-size of an EXPLICIT scheme cannot be big. They are limited by the stability criterion. If the time-step size is BIGGER than the limit GIVEN by this criterion, the results will GO extremely unstable.

The correct option is (a) 1000

Explanation: USING the forward DIFFERENCE method,

\(\FRAC{\PARTIAL U}{\partial r}=\frac{u_2-u_1}{\Delta r}=\frac{5-0}{0.5×10^{-2}}=\frac{5}{5×10^{-3}}=1000.\)

The correct ANSWER is (b) \(\frac{u_{i,j+1}-2u_{i,j}+u_{i,j-1}}{(\Delta y)^2}\)

The explanation: To get the second difference,

\(u_{i,j+1}+u_{i,j-1}=2 u_{i,j}+(\frac{\PARTIAL^2 U}{\partial y^2})_{i,j}(\Delta y)^2+⋯\)

\((\frac{\partial^2 u}{\partial y^2})_{i,j}=\frac{u_{i,j+1}-2 u_{i,j}+u_{i,j-1}}{(\Delta y)^2} +⋯\)

After TRUNCATING,

\((\frac{\partial^2 u}{\partial y^2})_{i,j}=\frac{u_{i,j+1}-2 u_{i,j}+u_{i,j-1}}{(\Delta y)^2}\).

Correct answer is (c) refining the grid

Easy EXPLANATION: It is not possible to DECREASE the discretization error as it will lead to increased ALGEBRA and COMPUTATION. So, the information about the discretization error can be used only for refining the girds and achieve the same level of discretization everywhere in the solution.

Right option is (c) It is simple to solve

The explanation is: Lax-Wendroff TECHNIQUE USES the finite difference method to get time-dependent solutions. The need for linearization DEPENDS upon the equation to be solved. Simultaneous equations are not required as the resulting SYSTEM is explicit. But the system is not simple to solve. It involves lengthy algebra to get the second order TERMS.

The correct option is (d) second-ORDER

Easiest EXPLANATION: The Lax-Wendroff technique is second order accurate in both space and time. The FIRST term in the truncation error has an order 2. This order of accuracy makes the algebra BEHIND the technique complex.

The correct option is (a) explicit, finite-DIFFERENCE METHOD

The BEST I can explain: Lax-Wendroff TECHNIQUE is particularly suitable for marching solutions of hyperbolic and parabolic partial differential equations. It is an explicit method which uses the finite difference scheme for marching solutions.

The CORRECT choice is (c) N LOG ⁡N

To explain I would say: The cost of COMPUTATION is reduced by FFT. FFT has a cost of computation of Nlog⁡ N orders. This is much lower than N^2, especially when N is large. This reduces the PROBLEM of computation cost in the Spectral method.

The correct ANSWER is (d) SECOND-order

The best explanation: The LEAST possible order of accuracy for the second derivatives is 2. There cannot be a first-order second derivative as the second derivatives need terms LESS than the second order for the approximation.

The CORRECT choice is (B) When the convergence is monotonic

Explanation: The RICHARDSON extrapolation is very useful as it is a simple method. But, it can give ACCURATE RESULTS only when the convergence is monotonic. This convergence represents the convergence of error while refining the grid.

The correct choice is (a) Difference between the solutions OBTAINED from systematically REFINED grids

To elaborate: Discretization error is originally the difference between the exact SOLUTION of the partial DIFFERENTIAL equation and the solution of the ALGEBRAIC equation. But, as the exact solution is not known, we estimate the discretization error as the difference between the solutions obtained from systematically refined grids.

Correct OPTION is (b) Grids should be uniform and FUNCTION must be periodic

Explanation: To get the full advantages of this Spectral METHOD, the function must be periodic of the DEPENDENT variable and the grids should be uniform. This makes the Spectral method inflexible when compared to the other discretization methods.

Correct choice is (a) TRUNCATION error is less

Explanation: As the time-step SIZE is very large, the truncation error may become large and the accuracy of RESULTS may be less when compared to that of the explicit SCHEMES. The total time of COMPUTATION is less. But the algorithm is difficult to set-up.

The CORRECT option is (a) Add the error ESTIMATE to the results of the FINEST grid

For EXPLANATION: When we have the results of many grid arrangements ranging from coarse to fine. A solution which is more accurate than the solution of the finest grid can be obtained by ADDING the error estimate to the results of the finest grid available.

The CORRECT choice is (b) (Δ t)^0,(Δ t)^1 and (Δ t)^2

The explanation is: The first three terms of the Taylor SERIES EXPANSION for the time marching TERM is used in the Lax-Wendroff Technique. This leads to the second-order accuracy of the system. Known VALUES at previous time-step are used to find the value at the current time-step.

Correct option is (a) FOURIER series

For explanation: Finite Difference METHODS use Taylor series. In spectral methods, spatial derivatives are evaluated using the Fourier series or ONE of their generalization. The SIMPLEST spectral methods deal with periodic FUNCTIONS specified by their values at a uniformly spaced set of points.

Correct answer is (b) N^2

Easiest EXPLANATION: The computational cost REQUIRED for computing FOURIER coefficients, if done in the most obvious MANNER, is N^2. This is prohibitively expensive. It is twice that of the backward substitution for Gauss-Elimination method.

Correct option is (c) Erroneous solutions may converge

To explain: The order of convergence is valid only when the convergence is monotonic. This is because, for two consecutive grids, RESULTS may ALSO converge even if the error is large. Then, a third grid ARRANGEMENT must be used to ASSURE the real convergence of the SOLUTION.

Correct CHOICE is (d) ARITHMETIC MEAN of \((\frac{\partial\rho}{\partial t})_{i,j}^t and (\frac{\partial\rho}{\partial t})_{i,j}^{t+\Delta t}\)

Explanation: The VALUE of \((\frac{\partial \rho}{\partial t})_{i,j}^{av}\) is the arithmetic mean of \((\frac{\partial \rho}{\partial t})_{i,j}\) at t and \((\frac{\partial \rho}{\partial t})_{i,j}\) at t+Δt.

\((\frac{\partial \rho}{\partial t})_{i,j}^{av}=\frac{1}{2}[(\frac{\partial\rho}{\partial t})_{i,j}^t+(\frac{\partial\rho}{\partial t})_{i,j}^{t+\Delta t}]\).

Correct ANSWER is (B) Differentiate the continuity equation with RESPECT to time

The explanation: Differentiating the value of any variable or the value of its DERIVATIVE have no sense as it will result in zero. To differentiateup \(\rho_{i,j}^t\) with respect to time twice, the equation for \(\rho_{i,j}^t\) should be known. But, it is not. So, differentiating the continuity equation with respect to time is the only way. REMEMBER the continuity equation gives \((\frac{\partial \rho}{\partial t})\).

Correct choice is (a) as the grid size is REDUCED, the approximations converge to the exact solution with an error proportional to m powers of the grid size

Easiest EXPLANATION: An order of ACCURACY m means that the TRUNCATION error starts with a TERM proportional to grid size^m. So, the statement “as the grid size is reduced, the approximations converge to the exact solution with an error proportional to m powers of the grid size” is correct.

The correct option is (c) \(\frac{u_{i+1,J}-u_{i-1,j}}{2\Delta x}\)

For explanation: The only second-order accurate FINITE difference APPROXIMATION of the first derivative is the central difference. For GETTING the central difference term,

\(u_{i+1,j}=u_{i,j}+(\frac{\partial u}{\partial x})_{i,j}\Delta x+(\frac{\partial ^2 u}{\partial x^2})_{i,j}\frac{(\Delta x)^2}{2}+⋯\)

\(u_{i-1,j}=u_{i,j}-(\frac{\partial u}{\partial x})_{i,j}\Delta x+(\frac{\partial ^2 u}{\partial x^2})_{i,j}\frac{(\Delta x)^2}{2}+⋯\)

To get \((\frac{\partial u}{\partial x})_{i,j^,}\)

\(u_{i+1,j}-u_{i-1,j}=2(\frac{\partial u}{\partial x})_{i,j} \Delta x+⋯\)

\((\frac{\partial u}{\partial x})_{i,j}=\frac{u_{i+1,j}-u_{i-1,j}}{2 \Delta x}\) .

The correct choice is (B) discretization ERROR

To explain I would say: The difference between the exact solution of the PARTIAL differential equation and the solution of the algebraic equation is the discretization error. Mathematically

Φ=Φnum+∈d

Where,

Φ → Exact solution

Φnum → Numerical solution

∈d → Discretization error

Correct choice is (c) TRUNCATION error

Explanation: Discretization error OCCURS because of the truncation errors which ARISE while discretizing the PDES. It is NAMED truncation error as the root cause of it is the truncation of the higher order terms in the series expansion.

Correct ANSWER is (b) Explicit linear form

Best explanation: The one-dimensional heat CONDUCTION equation is

\(\frac{\partial T}{\partial t}=α \frac{\partial^2 T}{\partial t^2} \)

Applying the difference APPROXIMATIONS,

\(\frac{T_i^{n+1}-T_i^n}{\DELTA t}=\ALPHA \frac{T_{i+1}^n-2T_i^n+T_{i-1}^n}{(\Delta x^2)} \)

\(T_i^{n+1}=T_i^n+\alpha\frac{\Delta t(T_{i+1}^n-2T_i^n+T_{i-1}^n)}{(\Delta x^2)} \)

The equation is in explicit linear form.

Right answer is (b) 2

To explain: The first term in the truncation error of the central difference for the MIXED DERIVATIVE \(\frac{\partial^2 U}{\partial x\partial y} \,is\, -(\frac{\partial^4 u}{\partial x^3 \partial y})\frac{(\Delta x)^2}{12}\). So, the order of ACCURACY is 2.

The CORRECT OPTION is (a) 1

Explanation: The lowest order term in the truncation error given is \(\frac{\Delta x}{2}\), which is of the order 1. This defines the order of ACCURACY of the equation. So, the order of accuracy here is 1.

Correct CHOICE is (c) (Δ t)^0 and (Δ t)^1

For EXPLANATION: Only the first TWO terms in the Taylor series expansion is used in the MacCormack’s technique. The first two terms are (Δ t)^0 and (Δ t)^1. All other higher-order terms are omitted. But, the order of accuracy is maintained here as two.

Right answer is (d) Second-order

Easiest explanation: MacCormack’s TECHNIQUE is second order ACCURATE in both space and TIME. There is a SPECIAL method used in MacCormack’s technique to make the order of accuracy two, even after reducing the lengthy ALGEBRA.

The correct option is (c) x-momentum EQUATION

Explanation: The x-momentum equation gives the time derivative if the x-component of velocity at a particular time in TERMS of the other flow VARIABLES and their special derivatives. So, this can be used to GET the time derivative \((\frac{\partial U}{\partial t})_{i,j}^t\).

The correct CHOICE is (d) \((\FRAC{\partial ^2 \RHO}{\partial t^2 })_{i,j}^t\)

The best I can explain: The second order term in the Taylor series expansion of the Lax-Wendroff technique is its disadvantage. This term leads to a complex ALGEBRA while getting it using the difference schemes. The LENGTHY algebra here is the only considerable disadvantage of this technique.

Right answer is (a) can be easily generated

Explanation: Spectral METHOD can easily be generated for HIGHER DERIVATIVES. The FOURIER coefficients will vary in higher order derivatives. Other than this, there are not much changes NEEDED for higher orders.

Right option is (d) Crank-Nicolson scheme

The best explanation: Crank-Nicolson scheme is used to SOLVE problems governed by PARABOLIC equations. They result in implicit time-dependent problems. In CFD, they are usually used for finite difference SOLUTIONS of boundary layer problems.

Correct option is (c) Non-linear equations

To explain: Though the implicit SCHEME has a great advantage of larger time steps, each step in an implicit scheme is large and TAKES more COMPUTATIONAL time. If the equation is non-linear, solving the system simultaneously will become more difficult. USUALLY, for these cases, the equations are LINEARIZED.

The correct answer is (a) FFT

The explanation: FFT STANDS for Fast Fourier Transform which is an algorithm for finding DISCRETE Fourier Transform (DFT) of a sequence or the Inverse Discrete Fourier Transform (IDFT). This is used to REDUCE the computational COST.

The correct choice is (d) To INCREASE the rate of convergence

To elaborate: RICHARDSON EXTRAPOLATION is a sequence acceleration METHOD. It is used to increase the rate of convergence of a SYSTEM. In the finite difference method, it is used to find accurate results from the discretized results.

The CORRECT choice is (b) the exact PARTIAL differential equation and the discretized equations

The EXPLANATION: Truncation ERROR is the difference between the exact partial differential equation and the discretized ALGEBRAIC equation. This arises as we cut-off the higher order terms in the Taylor series expansion.

Correct OPTION is (d) Stability

Best explanation: Implicit schemes do not have any restriction for the time-step size. They are stable for LARGE time-steps also. Some of the implicit schemes are EVEN UNCONDITIONALLY stable. Stability problems do not arise in implicit schemes.

The correct option is (d) \(\frac{u_{i+1,j}-u_{i,j}}{\Delta x}\)

The best I can explain: To GET the first-order FORWARD DIFFERENCE APPROXIMATION,

The Taylor series expansion of ui+1,j is

\(u_{i+1,j}=u_{i,j}+(\frac{\partial u}{\partial x})_{i,j}\Delta x+(\frac{\partial ^2 u}{\partial x^2})_{i,j}\frac{(\Delta x)^2}{2}+⋯\)

\((\frac{\partial u}{\partial x})_{i,j}=\frac{u_{i+1,j}-u_{i,j}}{\Delta x}\).