Order of accuracy m means _____________

as the grid size is reduced, the approximations converge to the exact solution with an error proportional to m times of the grid size

as the grid size is reduced, the approximations converge to the exact solution with an error proportional to m powers of the grid size

as the grid size is reduced, the approximations converge to the exact solution with an error proportional to m times of the grid size

as the grid size is reduced, the approximations diverge from the exact solution with an error proportional to m powers of the grid size

as the grid size is reduced, the approximations diverge from the exact solution with an error proportional to m times of the grid size

Find the central second difference of u in y-direction using the Taylor series expansion.

\(\frac{u_{i,j+1}-2u_{i,j}-u_{i,j-1}}{(\Delta y)^2}\)

\(\frac{u_{i,j+1}+2u_{i,j}+u_{i,j-1}}{(\Delta y)^2}\)

\(\frac{u_{i,j+1}-2u_{i,j}+u_{i,j-1}}{(\Delta y)^2}\)

\(\frac{u_{i,j+1}-2u_{i,j}-u_{i,j-1}}{(\Delta y)^2}\)

\(\frac{u_{i,j+1}+2u_{i,j}-u_{i,j-1}}{(\Delta y)^2}\)

Using the Taylor series expansion, What is the first term of the truncation error of the finite difference equation \((\frac{\partial u}{\partial x})_{i,j}=\frac{u_{i+1,j}-u_{i,j}}{\Delta y}\)?

\((\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{3}\)

\(-(\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{2}\)

\((\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{3}\)

\(-(\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{3}\)

\((\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{2}\)

Find the first-order forward difference approximation of \((\frac{\partial u}{\partial x})_{i,j}\) using the Taylor series expansion.

\(\frac{u_{i,j+1}-u_{i,j}}{2 \Delta x}\)

\(\frac{u_{i+1,j}-u_{i,j}}{2 \Delta x}\)

\(\frac{u_{i,j+1}-u_{i,j}}{\Delta x}\)

\(\frac{u_{i+1,j}-u_{i,j}}{\Delta x}\)

Find the second-order accurate finite difference approximation of the first derivative of the velocity component (u) in the x-direction using the Taylor series expansion. (Note: i and j are in the x and y-direction respectively).

\(\frac{u_{i,j+1}-u_{i,j-1}}{2\Delta x}\)

\(\frac{u_{i,j+1}-u_{i,j-1}}{\Delta x}\)

\(\frac{u_{i+1,j}-u_{i-1,j}}{\Delta x}\)

\(\frac{u_{i+1,j}-u_{i-1,j}}{2\Delta x}\)

\(\frac{u_{i,j+1}-u_{i,j-1}}{2\Delta x}\)

Consider the equation \((\frac{\partial u}{\partial y})_{i,j}=(\frac{u_{i,j}-u_{i,j-1}}{\Delta y})\) formulated using the Taylor series expansion. Find the type of equation.

second-order forward difference

first-order rearward difference

second-order forward difference

second-order rearward difference

10 + Mcqs in Finite Difference Method in Computational Fluid Dynamics Page 1 McqOptions

1.	Order of accuracy m means _____________
A.	as the grid size is reduced, the approximations converge to the exact solution with an error proportional to m powers of the grid size
B.	as the grid size is reduced, the approximations converge to the exact solution with an error proportional to m times of the grid size
C.	as the grid size is reduced, the approximations diverge from the exact solution with an error proportional to m powers of the grid size
D.	as the grid size is reduced, the approximations diverge from the exact solution with an error proportional to m times of the grid size
Answer» B. as the grid size is reduced, the approximations converge to the exact solution with an error proportional to m times of the grid size

Discussion

2.	What is the least order of accuracy for the second derivatives?
A.	first-order
B.	third-order
C.	fourth-order
D.	second-order
Answer» E.

Discussion

3.	Find \(\frac{\partial u}{\partial r}\) at point 1 using forward difference method.
A.	1000
B.	100
C.	500
D.	5000
Answer» B. 100

Discussion

4.	What is the order of the central difference for the mixed derivative \(\frac{\partial^2 u}{\partial x\partial y}\) while approximated using the Taylor series expansion?
A.	1
B.	2
C.	3
D.	4
Answer» C. 3

Discussion

5.	Find the central second difference of u in y-direction using the Taylor series expansion.
A.	\(\frac{u_{i,j+1}+2u_{i,j}+u_{i,j-1}}{(\Delta y)^2}\)
B.	\(\frac{u_{i,j+1}-2u_{i,j}+u_{i,j-1}}{(\Delta y)^2}\)
C.	\(\frac{u_{i,j+1}-2u_{i,j}-u_{i,j-1}}{(\Delta y)^2}\)
D.	\(\frac{u_{i,j+1}+2u_{i,j}-u_{i,j-1}}{(\Delta y)^2}\)
Answer» C. \(\frac{u_{i,j+1}-2u_{i,j}-u_{i,j-1}}{(\Delta y)^2}\)

Discussion

6.	Using the Taylor series expansion, What is the first term of the truncation error of the finite difference equation \((\frac{\partial u}{\partial x})_{i,j}=\frac{u_{i+1,j}-u_{i,j}}{\Delta y}\)?
A.	\(-(\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{2}\)
B.	\((\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{3}\)
C.	\(-(\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{3}\)
D.	\((\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{2}\)
Answer» B. \((\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{3}\)

Discussion

7.	Find the first-order forward difference approximation of \((\frac{\partial u}{\partial x})_{i,j}\) using the Taylor series expansion.
A.	\(\frac{u_{i,j+1}-u_{i,j}}{2 \Delta x}\)
B.	\(\frac{u_{i+1,j}-u_{i,j}}{2 \Delta x}\)
C.	\(\frac{u_{i,j+1}-u_{i,j}}{\Delta x}\)
D.	\(\frac{u_{i+1,j}-u_{i,j}}{\Delta x}\)
Answer» E.

Discussion

8.	Find the second-order accurate finite difference approximation of the first derivative of the velocity component (u) in the x-direction using the Taylor series expansion. (Note: i and j are in the x and y-direction respectively).
A.	\(\frac{u_{i,j+1}-u_{i,j-1}}{\Delta x}\)
B.	\(\frac{u_{i+1,j}-u_{i-1,j}}{\Delta x}\)
C.	\(\frac{u_{i+1,j}-u_{i-1,j}}{2\Delta x}\)
D.	\(\frac{u_{i,j+1}-u_{i,j-1}}{2\Delta x}\)
Answer» D. \(\frac{u_{i,j+1}-u_{i,j-1}}{2\Delta x}\)

Discussion

9.	Consider the equation \((\frac{\partial u}{\partial y})_{i,j}=(\frac{u_{i,j}-u_{i,j-1}}{\Delta y})\) formulated using the Taylor series expansion. Find the type of equation.
A.	first-order forward difference
B.	first-order rearward difference
C.	second-order forward difference
D.	second-order rearward difference
Answer» C. second-order forward difference

Discussion

10.	The truncation error in a finite difference expansion is \(-(\frac{\partial^2 u}{\partial x^2})_{i,j} \frac{\Delta x}{2}-(\frac{\partial^3 u}{\partial x^3})_{i,j} \frac{(\Delta x)^3}{6}\). What is the order of accuracy of the finite difference equation?
A.	1
B.	2
C.	-2
D.	-1
Answer» B. 2

Discussion

Explore topic-wise MCQs in Computational Fluid Dynamics.

Order of accuracy m means _____________

What is the least order of accuracy for the second derivatives?

Find \(\frac{\partial u}{\partial r}\) at point 1 using forward difference method.

What is the order of the central difference for the mixed derivative \(\frac{\partial^2 u}{\partial x\partial y}\) while approximated using the Taylor series expansion?

Find the central second difference of u in y-direction using the Taylor series expansion.

Using the Taylor series expansion, What is the first term of the truncation error of the finite difference equation \((\frac{\partial u}{\partial x})_{i,j}=\frac{u_{i+1,j}-u_{i,j}}{\Delta y}\)?

Find the first-order forward difference approximation of \((\frac{\partial u}{\partial x})_{i,j}\) using the Taylor series expansion.

Find the second-order accurate finite difference approximation of the first derivative of the velocity component (u) in the x-direction using the Taylor series expansion. (Note: i and j are in the x and y-direction respectively).

Consider the equation \((\frac{\partial u}{\partial y})_{i,j}=(\frac{u_{i,j}-u_{i,j-1}}{\Delta y})\) formulated using the Taylor series expansion. Find the type of equation.

The truncation error in a finite difference expansion is \(-(\frac{\partial^2 u}{\partial x^2})_{i,j} \frac{\Delta x}{2}-(\frac{\partial^3 u}{\partial x^3})_{i,j} \frac{(\Delta x)^3}{6}\). What is the order of accuracy of the finite difference equation?