Computational Fluid Dynamics Multiple Choice Questions on “Transient Flows – Euler Schemes”.
1. Which of these equations give the derivative of the function T at time t as given by the Crank-Nicolson scheme?
a) (frac{T(t+Delta t)-T(t-Delta t)}{2 Delta t})
b) (frac{T(t+Delta t)+T(t-Delta t)}{2 Delta t})
c) (frac{T(t+Delta t)-T(t-Delta t)}{Delta t})
d) (frac{T(t+Delta t)+T(t-Delta t)}{Delta t})
Answer: a
Clarification: The Crank-Nicolson scheme uses the previous and the next steps to get the derivative at the current step. Expressing it mathematically,
(frac{partial T(t)}{partial t} = frac{T(t+Delta t)-T(t-Delta t)}{2 Delta t}).
2. The approximation of the derivative taken by the Crank-Nicolson scheme is the same as the __________ of spatial derivative.
a) second order forward difference approximation
b) backward difference approximation
c) forward difference approximation
d) central difference approximation
Answer: d
Clarification: The central difference scheme of the spatial derivative uses the previous and the next neighbours for the approximation. The Crank-Nicolson scheme also uses such an approximation for its time derivative.
3. The Crank-Nicolson scheme is ________
a) fourth-order accurate
b) third-order accurate
c) second-order accurate
d) first-order accurate
Answer: c
Clarification: The order of accuracy of the Crank-Nicolson scheme is two. It has better accuracy than the forward Euler scheme. This can be obtained using the Taylor series expansion of the temporal derivative.
4. To find the values at the current time-step, the Crank-Nicolson scheme uses ___________
a) t-Δt and t+Δ t steps
b) t-Δ t and t-2Δ t steps
c) t+Δ t and t+2Δ t steps
d) t and t+Δt steps
Answer: b
Clarification: Like the forward Euler scheme, the Crank-Nicolson scheme also uses the older steps only to get the values at the current node. It uses the values at the previous step and at the step previous to it.
5. For the transient convection problems, the Crank-Nicolson scheme is stable when _________
a) CFLconv≤2
b) CFLconv≤1
c) CFLconv≥-2
d) CFLconv≥-1
Answer: a
Clarification: The Crank-Nicolson scheme is not stable always. It is only conditionally stable. For the transient convection problems, the scheme is stable only when CFLconv≤2.
6. For which of these problems is the Crank-Nicolson scheme unconditionally stable?
a) Compressible flows
b) Advection problems
c) Diffusion problems
d) Convection-Diffusion problems
Answer: c
Clarification: When the Crank-Nicolson scheme is applied to the diffusion problems, there is no restriction to the time-step from stability side. It is unconditionally stable for this case. This is why the scheme is often used for diffusion problems.
7. According to the Adams-Moulton scheme, the derivative of a function T at time-step t is given by _________
a) (frac{3 T(t)+4T(t-Delta t)-T(t-2Delta t)}{2Delta t})
b) (frac{3 T(t)-4T(t-Delta t)-T(t-2Delta t)}{2Delta t})
c) (frac{3 T(t)+4T(t-Delta t)+T(t-2Delta t)}{2Delta t})
d) (frac{3 T(t)-4T(t-Delta t)+T(t-2Delta t)}{2Delta t})
Answer: d
Clarification: To find the derivative, the Adams-Moulton method uses the previous and the second previous steps. The mathematical expression is
(frac{partial T(t)}{partial t}=frac{3 T(t)-4T(t-Delta t)+T(t-2Delta t)}{2Delta t}).
8. The Adams-Moulton scheme is __________
a) explicit
b) implicit
c) a two-level scheme
d) a three-level scheme