Spatial and Temporal Discretization

Question 1

What are the three fundamentals of CFD?

Answer 1

• Navier Stokes Equations
• Finite Volume Methods
• Spatial and Temporal Discretization

Question 2

What three important Pre-Processing Concepts?

Answer 2

• Boundary Conditions
• Mesh Generation
• Turbulence Models

Question 3

What are two important Post Processing Concepts?

Answer 3

• Visualization
• Analysis

Question 4

What is Method of Lines and what are its benifits?

Answer 4

• Separate discretization in space and time
• Largest flexibility
• Each term can be treated differently to yield different accuracies

Question 5

What are the three properties of central differencing?

Answer 5

• The variation between centroids is linear
• Second rate accurate
• Unbounded

Question 6

What is Upwind differenceing?

Answer 6

• Depends on direction of mass flux
• First order accurate
• Bounded
• Diffusive

Question 7

What is linear upwind differencing?

Answer 7

• We use gradient to improve accuracy of extrapolation
• Second order accurate
• Unbounded

Question 8

What is the function of a flux limiter?

Answer 8

• To resolve all physical problems as precisely as possible
• To suppress any numerical oscillations

Question 9

What is total variation diminishing?

Answer 9

Is a measure of how a function changes across grid values

Question 10

What are two level methods for temporal discretization?

Answer 10

• Forward Euler
• Backward Euler
• Crank Nickelson

Question 11

What interpolation schemes are used for the convective flux and diffusive flux / pressure gradient?

Answer 11

Linear Upward Differencing / Central Differencing

Question 12

What kind of interpolation scheme is good for accuracy
/ stability

Answer 12

Linear/Central Differencing
/ Upward Differencing

Question 13

How do you switch between interpolation schemes?

Answer 13

Blending functions

Question 14

What problems are caused by Lower / Higher order schemes?

Answer 14

Dissipation / Fluctuation

Question 15

What does Godunov say?

Answer 15

Linear numerical schemes for solving PDEs having the property of not generating new extrema can be at most first order accurate,

Question 16

How do flux limiters work?

Answer 16

In non-monotonic regions, switch to upwind differencing, in monotonic regions, retain second order accuracy.

Question 17

What is total variation diminishing?

Answer 17

Measure of how a function changes across eg. grid values

Question 18

What are the properties of Minmod?

Answer 18

• Works well for all signals
• Quite dissipative

Question 19

What are the properties of superbee?

Answer 19

• Generally compressive
• Generates sharp gradients in smooth regions

Question 20

What are the properties of MUSCL?

Answer 20

• Midpoint between superbee and minmod

Question 21

What are the properties of two level methods in temporal discretization?

Answer 21

• At most second order accurate
• Only consider values from two different time steps

Question 22

What are the properties of Forward Euler?

Answer 22

• Evaluates at the existing time level
• Explicit
• Error is first order
• Time step is restricted to the stability of the underlying solver
• used primarily to capture the transient behavior of moving waves

Question 23

What are the properties of Backward Euler?

Answer 23

• Evaluates at future time level
• Implicit
• Error is 2st order
• iteratively solved at each time level before moving to next step
• Unconditionally stable wrt time

Question 24

What are the properties of crank nickelson?

Answer 24

• evaluates at existing and future time level
• Implicit
• 2nd order accurate

Question 25

What are some higher order explicit methods?

Answer 25

- Adams - Runge- Kutta

Question 26

What are some higher order Implicit methods?

Answer 26

LU-SGS

Question 27

What behavior does the ratio of successive gradients have in monotonic / non-monotonic regions.

Answer 27

Non-Monotonic: r < 0 Monotonic: r = 1 for constant gradients 0 < r < 1 for linear gradients r > 1 non-linear gradients

Question 28

What are the seven discretization schemes mentioned on the lecture?

Answer 28

- Linear/ Central Differencing - Upwind Differencing - Linear Upwind differencing - Flux limiter schemes - Flux-vector splitting schemes - Flux-differencing splitting schemes - ENO/WENO

Question 29

What is the CFL condition?

Answer 29

The time step has to be less than or equal to cell size / flow velocity

Question 30

What is the formula for central differencing?

Answer 30

\phi_f =\psi * \phi_D + (1-\psi) * \phi_C \psi = (x_f - x_c) / (|x_D - x_C|)

Question 31

What is the formula for upwind differencing?

Answer 31

F_f = \rho_f (u_f . n) S_f