Engineering Questions with Answers - Multiple Choice Questions
Computational Fluid Dynamics – Numerical Methods – Coupled Equations and Non-Linear Equations Solution
1 - Question
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
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 - Question
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
Answer: c 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 - Question
For solving for a single unknown in sequential solvers ____________ is used. a) Direct solver b) LU decomposition c) Elimination method d) Iterative solver
Answer: d 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 - Question
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
Answer: b 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 - Question
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
Answer: a 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 - Question
Newton’s method linearizes the function using ___________ a) McLaurin series b) Laurent series c) Taylor series d) Fourier series
Answer: c 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 - Question
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
Answer: b 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 - Question
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
Answer: d 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 - Question
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
Answer: c 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 - Question
While using the Picard iteration, how is the source term decomposed? a) qΦ=b0+b1 Φ b) qΦ=b0 Φ+b1 Φ2 c) qΦ=b0 Φ d) qΦ=b0+b1 Φ2
Answer: a Explanation: Picard iteration is used with the source term to decompose and linearize it. It decomposes to qΦ=b0+b1 Φ. The term b0 is absorbed by the RHS of the system. The term b1 Φ is added to the coefficient matrix.