This set of Computational Fluid Dynamics Multiple Choice Questions & Answers (MCQs) focuses on “Numerical Methods – Discretization Approaches”.
1. The diagram represents a one-dimensional mesh.
The conservation equations are applied to which of these points while discretizing the equation using the Finite Difference Method?
a) 5, 7
b) 2, 3
c) 5, 6, 7
d) 1, 2, 3, 4
Explanation: The Finite Difference Method applies conservation equation to the nodes of each cell. The vertices of the cells form the nodes. So, points 1, 2, 3 and 4 are used for discretizing.
2. Which of these is the oldest method for numerical solution of partial differential equations?
a) Finite Element Method
b) Finite Difference Method
c) Finite Volume method
d) Spectral Element Method
Explanation: The Finite Difference Method is the oldest method for solving partial differential equations numerically. It is believed that this method is developed by Euler in the 18th century. This is the easiest method too.
3. Which of these properties is not ensured in Finite Difference Methods?
Explanation: Though the Finite Difference method is the easiest, it is suitable only for simple problems involving structured grids. Another main disadvantage is that conservation is not enforced unless special care is taken.
4. To obtain the derivatives of the variables with respect to the coordinates, which of these approximations are used in the Finite Difference Method?
a) Taylor series and polynomial fitting
b) Fourier series and polynomial fitting
c) Taylor series and Fourier series
d) Taylor series and interpolation
Explanation: The conservation equation of the partial differential form is used by the Finite Difference Method. To approximate the derivatives in this equation, Taylor series and polynomial fitting are used.
5. Which of these is an advantage of the Finite Difference Method over the Finite Volume Method?
d) Complex problems
Explanation: The Finite Volume Method cannot be applied to higher orders. The disadvantage of the Finite Volume Method, when compared to the Finite Difference Method, is that for orders higher than second order are more difficult to develop in 3-D.
6. The diagram represents a one-dimensional mesh.
The conservation equations are applied to which of these points while discretizing the equation using the Finite Volume Method?
a) 5, 7
b) 2, 3
c) 5, 6, 7
d) 1, 2, 3, 4
Explanation: Governing equations are applied to each control volume at its centre in the Finite Volume Method. In the diagram, points 5, 6 and 7 represent cell centres. To get the nodal values, these answers are interpolated.
7. Conservation of Finite Volume Method depends on ___________
a) surface integrals
b) volume integrals
Explanation: The Finite Volume Methods are conservative as long as the surface integrals which represent the convective and diffusive fluxes are the same for two control volumes sharing the same boundary.
8. A hybrid method of which of these methods exists?
a) Finite Difference Method and Finite Element Method
b) Finite Volume Method and Finite Difference Method
c) Finite Volume Method and Finite Element Method
d) Finite Volume Method and Spectral Element Method
Explanation: A hybrid method integrating Finite Volume Method and Finite Element Method called the Control Volume based Finite Element Method (CV-FEM) is also used for solving PDEs. In this, shape functions are used to describe the variation of the variables over an element.
9. In which of these methods, the calculation for interior nodes is done at the post-processing stage?
a) Spectral element method
b) Boundary element method
c) Finite Element method
d) Finite Volume Method
Explanation: In Boundary element method, the given boundary conditions are used to fit the boundary values into the integral equations. After this, using the boundary solutions, the solution for the interior nodes is carried out at the post-processing stage.
10. Which of these statements are wrong considering Control Volume based Finite Element Method?
a) Control volumes are formed
b) Integral form of the conservation equation is used
c) The centroid of the elements is used
d) Conservation equations are applied to the centroids
Explanation: In CV-FEM, control volumes are formed by joining the centroids of the elements. The conservation equation in the integral form is applied to these control volumes as in the Finite Volume Method.
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