Engineering Questions with Answers - Multiple Choice Questions

# Computational Fluid Dynamics – Finite Difference Methods – Lax-Wendroff Technique

1 - Question

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
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 - Question

What is the order of accuracy of the Lax-Wendroff technique?
a) fourth-order
b) third-order
c) first-order
d) second-order
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 - Question

Which series expansion is used by the Lax-Wendroff Technique?
a) Taylor Series
b) Fourier series
c) McLaurin series
d) Laurent series
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 - Question

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
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 - Question

Expand the term ρt+Δti,j 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) ρ(i,j)t+(∂ρ∂t)ti,jΔt+(∂2ρ∂t2)ti,j(Δt)22
b) ρt+Δti,j+(∂ρ∂t)t+Δti,jΔt+(∂2ρ∂t2)t+Δti,j(Δt)22
c) ρavi,j+(∂ρ∂t)avi,jΔt+(∂2ρ∂t2)avi,j(Δt)22
d) ρt−Δti,j+(∂ρ∂t)t−Δti,jΔt+(∂2ρ∂t2)t−Δti,j(Δt)22
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. ρt+Δti,j=ρ(i,j)t+(∂ρ∂t)ti,jΔt+(∂2ρ∂t2)ti,j(Δt)22

6 - Question

What is the disadvantage of the Lax-Wendroff technique?
a) Stability
b) Explicit
c) Order of accuracy
d) (∂2ρ∂t2)ti,j
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 - Question

Consider three-dimensional Euler equations. Which equation will you use to find the value (∂u∂t)ti,j?
a) Energy equation
b) y-momentum equation
c) x-momentum equation
d) Continuity equation
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 (∂u∂t)ti,j.

8 - Question

Consider three-dimensional Euler equations. What will you do to get the value of (∂2ρ∂t2)ti,j?
a) Differentiate ρti,j with respect to time twice
b) Differentiate the continuity equation with respect to time
c) Differentiate the value of (∂ρ∂t)ti,j with respect to time
d) Differentiate the value of ρti,j with respect to time twice
Explanation: Differentiating the value of any variable or the value of its derivative have no sense as it will result in zero. To differentiateup ρti,j with respect to time twice, the equation for ρti,j 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 (∂ρ∂t).

9 - Question

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
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 - Question

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