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Correct option is (a) truncation error

For EXPLANATION I WOULD say: When the grid sizes are sufficiently small, convergence is related to truncation error. The rate of convergence is governed by the ORDER of principal truncation error COMPONENT which is used to APPROXIMATE the partial differential equations.

The correct choice is (b) the results for TWO consecutive iterations are the same

To elaborate: Ideally, a solution of a SYSTEM of equations is SAID to be CONVERGED if the results of two consecutive iterations are exactly the same WITHOUT any variation. No more iterations are required after this.

The correct option is (a) Transient problems

The best EXPLANATION: For transient problems, a stable system keeps the error bounded when time INCREASES. STABILITY for a transient PROBLEM has SPECIAL characteristics. Here, stability guarantees that the method gives a bounded solution if the exact solution is bounded.

The correct CHOICE is (c) the NUMERICAL solution approaches the exact solution when time step and grid spacing tends to zero

The explanation is: A solution to some discretized algebraic equation is said to be consistent if that solution approaches the exact solution of the PARTIAL DIFFERENTIAL equation when time step and grid spacing are very small.

Correct choice is (C) 0

For explanation I WOULD say: CONSISTENCY is defined when the time step or grid spacing approaches 0. When this happens the discretization ERROR becomes zero. When discretization error is zero, it means that the solution matches with the analytical solution.

Right ANSWER is (B) Forward

For explanation I WOULD say: To find the current VALUES of PI and Qi, the previous values Pi-1 and Qi-1 should be known. So, this is started from the first equation and done in forward order.

Right answer is (d) Scarborough criterion

Easiest explanation: A system of linear equations can be taken as STABLE if it satisfies the Scarborough criterion. Scarborough criterion gives a condition about the COEFFICIENT matrix of the ALGEBRAIC system. This criterion ALSO gives information about the boundedness of the PROBLEM.

Correct choice is (a) Elimination method

Easiest EXPLANATION: Convergence is a property of the iterative solvers used for solving the discretized system of EQUATIONS. It cannot be defined for direct solvers as they do not have repeated STEPS and SIMILAR answers.

The correct answer is (c) an ITERATIVE METHOD

Easy explanation: THOMAS algorithm solves a system of equations with non-repeated sequence of OPERATIONS. It is a direct method to solve the system without involving repeated iterations and converging solutions.

Right answer is (c) the ratio should TEND to ZERO

To explain I would say: We know that consistency is satisfied if discretization error is zero. When discretization error is the ratio of grid spacing to TIME STEP, then actually the ratio should be zero. But, this is not practically possible as the grid spacing cannot be zero. So, grid spacing and time step must be reduced in a way that the ratio tends to zero.

Correct answer is (a) some POWERS of time-step

Best explanation: When time-step and GRID SPACING APPROACHES zero, discretization error should APPROACH zero. For this, it should be some powers of time-step and/ or grid spacing. If it is a function, it may or may not become zero.

The CORRECT ANSWER is (c) the neighbouring elements

For explanation: The discretized EQUATION for a particular element connects it with its neighbouring elements in all the sides in general. The particular sides depend upon the PROBLEM TAKEN.

The correct CHOICE is (b) BASED on efficiency and accuracy

Easiest explanation: In practical, the results of TWO iterations does not exactly match with each other. The iterations are STOPPED when the solution REACHES some acceptable tolerance. This tolerance is decided in a way that it affects neither the accuracy of the solution nor its efficiency.

The correct choice is (c) the system can be solved for different initial and boundary CONDITIONS

The explanation: A stable system of algebraic equation means that the system can be solved for different boundary conditions and initial conditions to get the flow PROPERTIES. While VARYING the boundary conditions, the system should not become UNSOLVABLE.

The correct answer is (d) discretization error

To ELABORATE: A NUMERICAL SOLUTION’s consistency has a direct dependence on discretization error. Discretization error occurs because of the discretization of the continuous solution. If this error is big, the solution will not match with the EXACT continuous solution.

The CORRECT choice is (a) Discretized CELLS

Easiest EXPLANATION: Discretized equations are formed for each element OBTAINED after discretizing the DOMAIN. So, the number of discretized cells and that of the discretized equations will be the same.

The correct choice is (a) GRID is fine

The best explanation: For fine rids, non-conservative schemes can also give a consistent and stable solution. The ERRORS due to non-conservation are negligible. These errors BECOME appreciable only if the grid is coarse and the grid size is more.

Correct OPTION is (b) Stability

The BEST I can explain: Inconsistency PROBLEMS arise when we truncate higher ORDER terms. These approximations are consistent is the same order terms are truncated always. Though this condition is satisfied, it is a must for the system of equations to be stable to SATISFY consistency.

The correct answer is (a) CFL number

Easiest EXPLANATION: Courant number is OTHERWISE CALLED CFL number. It is EXPANDED as Courant-Friedrichs-Lewy number. It is named after Richard Courant, Kurt Friedrichs and Hans Lewy who FIRST formed this number.

Right choice is (d) Taylor series expansion

To explain: CONSISTENCY wants the numerical solution to be the same as the ANALYTICAL solution. The numerical approximations of partial DIFFERENTIAL equations are done using the Taylor series expansion. The higher order terms in this series are neglected. This causes a DIFFERENCE between the numerical and the analytical solution.

Right choice is (d) only tri-DIAGONAL matrices

To explain I would SAY: The other name of the Thomas ALGORITHM is Tri-diagonal matrix algorithm. Tri-diagonal matrices are matrices with non-zero elements in the MAIN diagonal and the diagonals above and below it.

Right choice is (b) The TOTAL source or SINK in the DOMAIN is equal to the net flux through the boundaries

To explain: The general CONDITION of equal and opposite fluxes cannot be APPLIED to a system with sources or sinks. For this case, the total source or sink in the domain should be equal to the net flux through the boundaries of the whole domain.

The correct option is (a) both the GLOBAL and local domains

The explanation is: We KNOW that the conservation laws govern the flows and the properties like energy and mass are conserved in the global domain. This must be APPLICABLE to the local domain also after discretization. Otherwise, the solution will be UNREALISTIC.

The correct answer is (C) integrates

Best explanation: In the finite volume method, the governing equations are integrated over each CELL to form a semi-discretized equation. Then, the variation of FLOW variables is approximated.

Right ANSWER is (b) Boundedness

Explanation: LAX Equivalence Theorem states that “For a well-posed linear initial value PROBLEM solved by the finite difference approximation which satisfies consistency condition, stability is the necessary and SUFFICIENT condition for CONVERGENCE”.

Correct answer is (b) the time-step

Easiest explanation: Stability of transient schemes vary for implicit and explicit PROBLEMS. Stability of explicit transient SCHEME is ENSURED by limiting the time-step SIZE. Stability of implicit transient scheme is improved by under-relaxing the equations.

Correct option is (d) Spectral element METHOD

For explanation: Spectral element method multiplies the differential EQUATION by a random test function and then integrates it over the ENTIRE DOMAIN.

Right answer is (b) Closed-form

The best explanation: Closed form continuous PDEs are converted into discretized equations suitable for numerical SOLUTIONS. Analytical solution of this closed form PDEs will give continuous equations which can be used to GET the flow VARIABLE at any desired point in the DOMAIN.

Correct CHOICE is (d) when to stop the iterations

Best explanation: Iterative PROCESSES start with an initial guessed answer. This CONVERGES into the correct result as the NUMBER of iterations increases. Convergence criterion says when to stop this repeated PROCESS with acceptable error.

Right answer is (c) non-linear SYSTEMS with boundary CONDITIONS

Easy explanation: While solving non-linear coupled equations with boundary conditions, analysing STABILITY is difficult. So, the stability of a system is usually analysed for linear problems without boundary conditions. For non-linear systems, we rely on EXPERIENCE to KNOW its stability.

Right option is (d) equal magnitude and opposite sign

The explanation is: Flux leaving the face of one ELEMENT should be the flux entering the neighbouring element. So, the FLUXES of two near-by ELEMENTS should have equal magnitude and opposite sign. This ENSURES the conservativeness of the system.

The CORRECT option is (c) When the variation falls below a CERTAIN acceptable range

The explanation is: In real, the variation between two consecutive iterations cannot be EXACTLY the same. The value of variation will be constantly decreasing. So, the solution is said to be CONVERGED when the range of variation is acceptable.

Correct CHOICE is (d) Von Neumann’s method

For explanation: Von Neumann’s method is a widely used method of analysing the stability of any mathematical system. SINCE CFD uses NUMERICAL methods to solve problems, Von Neumann’s method is APPLICABLE to the CFD schemes also.

The correct choice is (d) \(P_1=\frac{b_1}{a_1};Q_1=\frac{d_1}{a_1}\)

Best EXPLANATION: In GENERAL,

\(P_i=\frac{b_i}{a_i-c_i P_{i-1}};Q_i=\frac{c_i Q_{i-1}+d_i}{a_i-c_i P_{i-1}}\)

As c1=0,

\(P_1=\frac{b_1}{a_1};Q_1=\frac{d_1}{a_1}\).

Right OPTION is (C) Backwards

The best explanation: The LAST value of ΦN can be FOUND using QN. Using this, the previous value is found using the formula Φi=PiΦi+1+Qi. So, it is done backwards.

The correct answer is (a) Linear equations solver

Explanation: Using a discretization METHOD, the GOVERNING partial differential EQUATION are converted into a system of algebraic equations. These discretized equations are SOLVED using the Thomas ALGORITHM.