This set of Computational Fluid Dynamics Multiple Choice Questions & Answers (MCQs) focuses on “Discretization Aspects – Consistency”.
1. A solution to some algebraic equation is said to be consistent if _________
a) the error is bounded
b) the computation time is not prohibitive
c) the numerical solution approaches the exact solution when time step and grid spacing tends to zero
d) the solution does not change with further iterations
Explanation: 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.
2. Consistency of a numerical solution is directly associated with __________
c) iterative error
d) discretization error
Explanation: 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.
3. What is the other name for Courant number?
a) CFL number
b) Peclet number
c) Nusselt number
d) Scarborough number
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.
4. Courant number is applicable for __________
a) implicit transient schemes
b) explicit transient schemes
c) quadratic schemes
d) high-resolution schemes
Explanation: Courant number is used in explicit time integration methods involving numerical methods. It relates grid size and time steps of the explicit transient schemes. It cannot be applied for implicit transient schemes.
5. Consistency is defined when the discretization error approaches __________
Explanation: 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.
6. Consistency comes into the picture because of _________
a) McLaurin series expansion
b) Power series expansion
c) Fourier series expansion
d) Taylor series expansion
Explanation: 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.
7. For consistency to have some relationship with discretization error, the discretization error should be ____________
a) some powers of time-step
b) some powers of grid spacing
c) some powers of time-step and/ or grid spacing
d) some function of time-step and/ or grid spacing
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.
8. For the solution of a system of discretized equations with consistent approximations to be consistent, which of these conditions is necessary?
Explanation: 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.
9. If the discretization error is the ratio of grid spacing to time step, then for the system to be consistent, which of these is correct?
a) the ratio should be equal to one
b) the ratio should be equal to zero
c) the ratio should tend to zero
d) the ratio should be equal to negative one
Explanation: 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.
10. Consistency should be ensured ___________
a) at the interior nodes
b) in the global domain
c) at the boundary nodes
d) at each node
Explanation: The algebraic equations for the partial differential governing equations are formed at each node of the domain. For each of these algebraic equations, consistency should be defined. There is no exception for the boundary nodes or the interior nodes.
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