Explore topic-wise InterviewSolutions in .

The correct ANSWER is (d) An n^th ORDER Runge-Kutta method is more accurate than the n^th order MULTIPOINT method

The EXPLANATION is: When comparing the Runge-Kutta method and the multipoint method, even if the order of accuracy is the same, the Runge-Kutta method is more accurate. This is because the COEFFICIENT of the Runge-Kutta method is small.

Right OPTION is (b) EXPLICIT Euler method

For explanation I would say: The prediction step USES the FORWARD (explicit) Euler method. The formula for this method is given as

Φ^n+1*=Φ^n+f(t^n,Φ^n )Δt.

Here, ^*indicates that this answer is not the FINAL one. This predicted value is corrected later.

The correct option is (a) THIRD-order

The BEST explanation: When the terms in the discretized form of the transient term USING the second-order UPWIND Euler scheme is further EXPANDED with the Taylor series expansion, the dispersion term of the third order arises.

Correct choice is (c) three, four

Explanation: The PISO PROCEDURE is repeated at each time-step to calculate the velocity and PRESSURE values. This algorithm results in third-order TEMPORALLY ACCURATE pressure values and fourth-order temporally accurate momentum values.

Correct answer is (b) small TIME-steps

Explanation: The PISO algorithm is advantageous compared to the SIMPLE algorithm that they do not NEED an iterative process at each time step. But the DISADVANTAGE is that they need the time steps to be small ENOUGH to PRODUCE accurate results.

The correct CHOICE is (a) both the first and second pressure-correction equations

For explanation I would SAY: The transient PISO algorithm has two corrector steps and HENCE two pressure-correction equations. Both of these equations are affected by the transient TERM. The source TERMS of these equations are altered.

Right answer is (c) Temporal INITIAL value PROBLEMS in ODEs

To ELABORATE: 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.

Correct answer is (a) second-ORDER ACCURATE

The explanation is: 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.

The correct answer is (d) Crank-Nicolson method

Best 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.

The CORRECT choice is (a) \(\frac{T(t+\DELTA t)-T(t-\Delta t)}{2 \Delta t}\)

To explain: The Crank-Nicolson scheme uses the previous and the NEXT steps to get the DERIVATIVE at the current step. EXPRESSING it mathematically,

\(\frac{\partial T(t)}{\partial t} = \frac{T(t+\Delta t)-T(t-\Delta t)}{2 \Delta t}\).

Right choice is (b) COMPUTATIONAL cost

Explanation: Though the Runge-Kutta methods are advantageous in the ACCURACY and STABILITY perspectives, this comes at the cost of computational cost. Computationally, the Runge-Kutta methods are very EXPENSIVE when compared to the other methods.

Correct ANSWER is (b) forward EULER method

Best explanation: The second-order Runge-Kutta method includes two steps. The first STEP can be called a half-step predictor. This is based on the forward Euler method which is an explicit method of first-order ACCURACY.

The correct option is (a) The current central COEFFICIENT for the finite difference and the finite volume schemes are the same

For EXPLANATION I would SAY: For the finite difference APPROACH,

\(a_c=(\frac{1}{\Delta t}+\frac{1}{\Delta t+\Delta t^o})\rho_C V_C\)

For the finite volume approach, the central coefficient is

FluxC=\((\frac{1}{\Delta t}+\frac{1}{\Delta t+\Delta t^o})\rho_C V_C\)

These TWO are the same irrespective of the approach.

Right OPTION is (b) implicit

The best EXPLANATION: The Adams-Moulton scheme wants all of its equations to be SOLVED simultaneously. It is not a time MARCHING scheme. It is an implicit scheme. So, it is computationally more EXPENSIVE.

Right choice is (d) central difference approximation

Easiest explanation: The central difference scheme of the SPATIAL DERIVATIVE USES the previous and the next NEIGHBOURS for the approximation. The Crank-Nicolson scheme ALSO uses such an approximation for its time derivative.

Correct answer is (B) EXPRESS the face values in terms of the cell values

The explanation: While discretizing the transient TERM, the semi-discretized EQUATION contains the values at the cell faces. If these face values are expressed in terms of the cell values, the complete discretized form of the equation can be obtained.

The CORRECT choice is (a) The finite volume and finite difference SCHEMES do not yield equivalent ALGEBRAIC EQUATIONS

Best explanation: The finite volume and finite difference schemes yield equivalent algebraic equations when the transient grid is UNIFORM. If the grid is non-uniform, the algebraic equations are not the same.

Right choice is (a) \(\FRAC{1}{2}\)(ρC ΦC)^t+\(\frac{1}{2}\)(ρC ΦC)^t+Δ t

Explanation: The Crank-Nicolson scheme GIVES EQUAL weight to both the cells which share the face. The FORMULA is given by,

(ρC ΦC)^t+Δt/2=\(\frac{1}{2}\)(ρC ΦC)^t+\(\frac{1}{2}\)(ρC ΦC)^t+Δt.

Correct ANSWER is (b) t-Δ t and t-2Δ t steps

The explanation: Like the FORWARD EULER SCHEME, the Crank-Nicolson scheme also uses the older steps only to get the values at the current node. It uses the values at the previous step and at the step previous to it.

Correct choice is (d) MIDPOINT rule corrector

The best explanation: The SECOND step of the second-order Runge-Kutta method is the corrector step. For this correction, midpoint rule is USED. This step MAKES this Runge-Kutta method a second-order method.

Correct answer is (d) LAGRANGE POLYNOMIAL

Explanation: 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.

Right ANSWER is (c) t-Δ t and t-2Δ t

Best EXPLANATION: The second-order upwind Euler scheme uses TWO upwind nodes to APPROXIMATE the VALUES. The immediate upwind and the far upwind of t-\(\frac{\Delta t}{2}\) are t-Δ t and t-2Δ t. The vales at these two nodes are used for the approximation.

Correct choice is (a) First-order

To explain I would say: 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.

The correct option is (d) DISPERSIVE term

Easy EXPLANATION: By expanding the results of the Crank-Nicolson SCHEME USING the Taylor series EXPANSION, the scheme is proved to be a second-ordered scheme. The third-ordered term omitted here is a dispersive term causing instabilities.

The correct option is (c) Crank-Nicolson SCHEME

To explain: For all the second-order schemes, the INTERPOLATION profile has to be modified when the transient grid is non-uniform. But this is not the case when the Crank-Nicolson scheme is used. There is no CHANGE in the interpolation profile NEEDED here.

Correct answer is (d) \(\frac{3 T(t)-4T(t-\Delta t)+T(t-2\Delta t)}{2\Delta t}\)

Easiest explanation: To FIND the derivative, the Adams-Moulton METHOD USES the previous and the second previous steps. The mathematical expression is

\(\frac{\PARTIAL T(t)}{\partial t}=\frac{3 T(t)-4T(t-\Delta t)+T(t-2\Delta t)}{2\Delta t}\).

Right option is (a) Midpoint RULE

Easiest 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.

Correct option is (c) the SCHEME is first-order

Easiest EXPLANATION: Since the second-order schemes use a stencil with TWO time-steps in the same DIRECTION, only the second-order schemes are AFFECTED by the non-uniformity of the grids. The first-order schemes are not affected.

Correct answer is (B) IMPLICIT and explicit first-order EULER schemes

Best EXPLANATION: The finite volume approach USING the Crank-Nicolson scheme can be reformulated using the first-order implicit Euler scheme and then the explicit Euler scheme which is modified and used in the form of extrapolation.

Right answer is (a) the convection TERM

To explain: The finite volume approach for the discretization of the RATE of change per unit time is more similar to the discretization of convection term while doing a spatial discretization. The only difference is that the FIRST case is based on time and the SECOND is based on spatial coordinates.

The correct ANSWER is (C) Simpson’s rule

Explanation: The third step of the fourth-order Runge-Kutta METHOD uses midpoint rule to correct the VALUES and the last step uses Simpson’s rule. This renders a fourth-order ACCURACY to the Runge-Kutta method.

Correct option is (a) CFL NUMBER

For explanation: The Crank-Nicolson scheme is an explicit scheme which uses the values before the FACE and after the face to FIND the value at the face. Therefore, its STABILITY is constrained by the CFL or COURANT number.

The correct choice is (c) t+Δt

To explain: In the first-order IMPLICIT EULER scheme, the VALUES at the cell faces are approximated by the values at cell centres of the backward DIRECTION. Therefore, the value at t+\(\frac{\DELTA t}{2}\) is replaced by the value at t.

Correct choice is (C) Diffusion problems

To explain I would say: When the Crank-Nicolson SCHEME is applied to the diffusion problems, there is no RESTRICTION to the time-step from STABILITY side. It is unconditionally stable for this case. This is why the scheme is often used for diffusion problems.