This set of Computational Fluid Dynamics Question Paper focuses on “Numerical Methods – Coupled Equations and Non-Linear Equations Solution”.

1. When coupled equations are ___________ sequential solutions are used.

a) linear and highly coupled

b) non-linear and uncoupled

c) linear and uncoupled

d) non-linear and highly coupled

View Answer

Explanation: There are two ways to solve coupled equations – simultaneous and sequential. In the simultaneous methods, equations are solved together for the unknowns. The sequential methods are used to solve a highly coupled system with linear equations.

2. In sequential methods for solving coupled equations, except the variable for which the equations are solved, the other variables are treated as ___________

a) zeros

b) unknowns

c) known values

d) ones

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Explanation: In sequential methods of solving coupled equations, the variable for which the system is solved is treated as unknown. All other variables are treated as known values with some approximations.

3. For solving for a single unknown in sequential solvers ____________ is used.

a) Direct solver

b) LU decomposition

c) Elimination method

d) Iterative solver

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Explanation: For each inner iteration, one variable is unknown and all other variables are treated as known values. It is ineffective to solve this system accurately for one unknown. So, the iterative solvers are preferred to direct solvers in this case.

4. In solving non-linear systems, there is a trade-off between ___________ and ___________

a) speed and stability

b) speed and security

c) stability and convergence

d) stability and error

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Explanation: In solving the non-linear system, there are two methods – Newton’s method and Global method. Newton’s method is faster and the Global method is guaranteed not to diverge. So, there is always a trade-off between speed and security.

5. The master method for solving the non-linear system of equations is __________

a) Newton’s method

b) Global method

c) Jacobi method

d) Gradient method

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Explanation: Newton Raphson is the most widely used method for solving a non-linear system of equations. It is preferred in most of the cases as the rate of convergence is more. It converges fast.

6. Newton’s method linearizes the function using ___________

a) McLaurin series

b) Laurent series

c) Taylor series

d) Fourier series

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Explanation: Newton’s method uses the first two terms of Taylor’s series to linearize the non-linear system. This is further simplified to get the formula to be iterated and get the roots.

7. Which of these creates a problem in Newton’s method for solving non-linear system of equations?

a) Taylor series

b) Jacobian

c) Convergence

d) Speed

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Explanation: At each iteration of Newton’s method, Jacobian has to be evaluated for the method to be effective. Evaluation of the Jacobian with n2 elements at each step will be expensive. Moreover, a direct method of evaluating the Jacobian does not exist.

8. When evaluation of the derivative of the non-linear function is not possible, which method is used?

a) Newton’s method

b) Global method

c) Jacobi method

d) Secant method

View Answer

Explanation: An alternative to Newton’s method is the Secant method. This is much slower than Newton’s method. However, when the derivative of the function cannot be evaluated, this method is chosen as it does not involve any derivative.

9. The non-linear terms like convection and source terms in a system are linearized using __________

a) Iterative gradient method

b) Jacobi method

c) Picard iteration

d) Incomplete LU decomposition

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Explanation: The usual solution of non-linear coupled system is the sequential decoupled method. For this, the non-linear terms are linearized using the Picard iteration method. This is applied for convection and source terms of the equation.

10. While using the Picard iteration, how is the source term decomposed?

a) q_{Φ}=b_{0}+b_{1} Φ

b) q_{Φ}=b_{0} Φ+b_{1} Φ^{2}

c) q_{Φ}=b_{0} Φ

d) q_{Φ}=b_{0}+b_{1} Φ^{2}

View Answer

Explanation: Picard iteration is used with the source term to decompose and linearize it. It decomposes to q

_{Φ}=b

_{0}+b

_{1}Φ. The term b

_{0}is absorbed by the RHS of the system. The term b

_{1}Φ is added to the coefficient matrix.

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