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The CORRECT ANSWER is (c) ADDITIONAL points where data is already available

For explanation I would say: 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.

The correct CHOICE is (d) Φ^n+1=Φ^n+\(\frac{1}{2}\) [f(t^n,Φ^n )+f(t^n+1,Φ^n+1*)]Δt

Best explanation: 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

Φ^n+1=Φ^n+\(\frac{1}{2}\) [f(t^n,Φ^n )+f(t^n+1,Φ^n+1*)]Δt.

The CORRECT choice is (b) Backward Euler method

For explanation I would SAY: 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.

Right option is (b) It INVOLVES the terms at the next time-step

Easy explanation: The Adams-Moulton method involves the INFORMATION at the UPCOMING steps also. The ADVANTAGE of the Adams-Moulton method is that it NEEDS only n steps to get an order of accuracy equal to n+1.

The correct option is (c) product of the previous central coefficient and previous velocity

To explain: To the source TERM of the u-momentum equation, the term \(a_P^0 \, u_P^0\) is added. Where, \(a_P^0\) is the initial central coefficient and \(u_P^0\) is the initial velocity in u-direction. A SIMILAR term is added to the sources of the other momentum equations ALSO.

The correct choice is (b) only one evaluation of derivative per TIME step

Explanation: 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.

Right choice is (b) \(\big|\Delta t\frac{\partial F}{\partial\phi}\big|<2\)

To explain: The forward Euler method is conditionally STABLE. For this method to be stable, it needs the following condition to be SATISFIED.

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

When the FUNCTION f is real, this becomes

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

Correct answer is (B) No DIFFUSION term

The explanation is: The transient term discretization ALSO has similar properties like the convection term discretization. The second-order upwind EULER scheme does not have any numerical diffusion or anti-diffusion terms. These terms are present for the first-order SCHEMES only.

The correct choice is (d) Operator splitting

Easy explanation: The temporal accuracy of the PISO ALGORITHM depends on the DIFFERENCING scheme used. This includes the operator splitting TECHNIQUE used. Operator splitting uses different methods to calculate different variables.

Right CHOICE is (C) the courant number of convection is equal to one

The best I can explain: The numerical DIFFUSION term of the first-order implicit Euler SCHEME and numerical anti-diffusion term of the first-order explicit Euler scheme are equal in MAGNITUDE and opposite in sign when the courant number of convection is equal to one.

The CORRECT option is (a) EULER METHODS

Easy explanation: All the steps of the Runge-Kutta method use the two-level formulae for INITIAL value problems. The first step uses the forward Euler method and the SECOND step uses the backward Euler method. Collectively, we can say that these two steps use Euler methods.

The CORRECT choice is (c) Four steps

Best EXPLANATION: All the Runge-Kutta methods are of high orders. The fourth-order Runge-Kutta method is a method which uses four steps. These four steps include the predictor and the CORRECTOR steps.

The correct option is (c) implicit and explicit methods

Easiest explanation: 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.

Right option is (d) Second-order

The explanation is: 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.

Correct choice is (b) t

The best explanation: The VALUE at t-\(\frac{\Delta t}{2}\) is at the interface of two CELLS. One has the cell centre t and the other has the cell centre t-Δ t. The first-order explicit Euler scheme is downstream biased. THEREFORE, the value at t is taken to approximate the value at t-\(\frac{\Delta t}{2}\).

Right OPTION is (d) numerical diffusion

The BEST I can explain: As the TRANSIENT term behaves LIKE the convection term while discretizing, numerical diffusion is produced by the first-order implicit Euler schemes. The VALUE of the numerical diffusion can be obtained using the Taylor series expansion.

The CORRECT choice is (a) At the (n+0.5)^TH point

Explanation: The first two STEPS of the fourth-order Runge-Kutta method find the values at the (n+0.5)^th point. It does not DIRECTLY move to the next step. It finds the value at an intermediate point between the current and the next POINTS.