This set of Computational Fluid Dynamics Multiple Choice Questions & Answers (MCQs) focuses on “Finite Difference Methods – Lax-Wendroff Technique”.
1. The Lax-Wendroff technique is ____________
a) explicit, finite-difference method
b) implicit, finite-difference method
c) explicit, finite volume method
d) implicit, finite volume method
View Answer
Explanation: Lax-Wendroff technique is particularly suitable for marching solutions of hyperbolic and parabolic partial differential equations. It is an explicit method which uses the finite difference scheme for marching solutions.
2. What is the order of accuracy of the Lax-Wendroff technique?
a) fourth-order
b) third-order
c) first-order
d) second-order
View Answer
Explanation: The Lax-Wendroff technique is second order accurate in both space and time. The first term in the truncation error has an order 2. This order of accuracy makes the algebra behind the technique complex.
3. Which series expansion is used by the Lax-Wendroff Technique?
a) Taylor Series
b) Fourier series
c) McLaurin series
d) Laurent series
View Answer
Explanation: Lax-Wendroff technique uses the Taylor series expansion to approximate its time derivatives. This makes the technique marching in time in an explicit way. The number of terms used for this expansion decides the accuracy of this system.
4. How many terms of the Taylor series expansion is used in the Lax-Wendroff technique?
a) (Δ t)1 and (Δ t)2
b) (Δ t)0, (Δ t)1 and (Δ t)2
c) (Δ t)0 and (Δ t)1
d) (Δ t)0
View Answer
Explanation: The first three terms of the Taylor series expansion for the time marching term is used in the Lax-Wendroff Technique. This leads to the second-order accuracy of the system. Known values at previous time-step are used to find the value at the current time-step.
5. Expand the term \(\rho_{i,j}^{t+\Delta t}\) for Lax-Wendroff technique.
Note:
t → Current time-step
t+Δt → Next time-step
av → Average time-step between t and t+Δ t
t-Δ t → Previous time-step
a) \(\rho_(i,j)^t+(\frac{\partial ρ}{\partial t})_{i,j}^t \Delta t+(\frac{\partial ^2 ρ}{\partial t^2 })_{i,j}^t \frac{(\Delta t)^2}{2} \)
b) \(\rho_{i,j}^{t+\Delta t}+(\frac{\partial \rho}{\partial t})_{i,j}^{t+\Delta t} \Delta t+(\frac{\partial^2 ρ}{\partial t^2 })_{i,j}^{t+\Delta t}\frac{(\Delta t)^2}{2}\)
c) \(\rho_{i,j}^{av}+(\frac{\partial \rho}{\partial t})_{i,j}^{av} \Delta t+(\frac{\partial ^2 ρ}{\partial t^2 })_{i,j}^{av}\frac{(\Delta t)^2}{2}\)
d) \(\rho_{i,j}^{t-\Delta t}+(\frac{\partial \rho}{\partial t})_{i,j}^{t-\Delta t} \Delta t+(\frac{\partial^2 \rho}{\partial t^2})_{i,j}^{t-\Delta t}\frac{(\Delta t)^2}{2}\)
View Answer
Explanation: Lax-Wendroff technique uses the previous time-step values to get the current time-step values using the Taylor series expansion. The first three terms of the Taylor’s series expansion is used.
\(\rho_{i,j}^{t+\Delta t} = \rho_(i,j)^t+(\frac{\partial ρ}{\partial t})_{i,j}^t \Delta t+(\frac{\partial ^2 ρ}{\partial t^2 })_{i,j}^t \frac{(\Delta t)^2}{2} \)
.
6. What is the disadvantage of the Lax-Wendroff technique?
a) Stability
b) Explicit
c) Order of accuracy
d) \((\frac{\partial ^2 \rho}{\partial t^2 })_{i,j}^t\)
View Answer
Explanation: The second order term in the Taylor series expansion of the Lax-Wendroff technique is its disadvantage. This term leads to a complex algebra while getting it using the difference schemes. The lengthy algebra here is the only considerable disadvantage of this technique.
7. Consider three-dimensional Euler equations. Which equation will you use to find the value \((\frac{\partial u}{\partial t})_{i,j}^t\)?
a) Energy equation
b) y-momentum equation
c) x-momentum equation
d) Continuity equation
View Answer
Explanation: The x-momentum equation gives the time derivative if the x-component of velocity at a particular time in terms of the other flow variables and their special derivatives. So, this can be used to get the time derivative \((\frac{\partial u}{\partial t})_{i,j}^t\).
8. Consider three-dimensional Euler equations. What will you do to get the value of \((\frac{\partial^2 ρ}{\partial t^2})_{i,j}^t\)?
a) Differentiate \(\rho_{i,j}^t\) with respect to time twice
b) Differentiate the continuity equation with respect to time
c) Differentiate the value of \((\frac{\partial \rho}{\partial t})_{i,j}^t\) with respect to time
d) Differentiate the value of \(\rho_{i,j}^t\) with respect to time twice
View Answer
Explanation: Differentiating the value of any variable or the value of its derivative have no sense as it will result in zero. To differentiateup \(\rho_{i,j}^t\) with respect to time twice, the equation for \(\rho_{i,j}^t\) should be known. But, it is not. So, differentiating the continuity equation with respect to time is the only way. Remember the continuity equation gives \((\frac{\partial \rho}{\partial t})\).
9. Which of these is wrong for the Lax-Wendroff technique?
a) Linearization is needed
b) Simultaneous equations are not required
c) It is simple to solve
d) It uses the finite difference method
View Answer
Explanation: Lax-Wendroff technique uses the finite difference method to get time-dependent solutions. The need for linearization depends upon the equation to be solved. Simultaneous equations are not required as the resulting system is explicit. But the system is not simple to solve. It involves lengthy algebra to get the second order terms.
10. Which is the technique used to overcome the disadvantages of the Lax-Wendroff technique?
a) Upwind scheme
b) MacCormack’s technique
c) Downwind scheme
d) Richtmeyer method
View Answer
Explanation: To overcome the lengthy algorithm of the Lax-Wendroff technique, MacCormack’s technique is used which can produce results of the same order of accuracy with a simpler method which does not want the second-order derivative.
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