Spatial and Temporal Discretization Flashcards
What are the three fundamentals of CFD?
- Navier Stokes Equations
- Finite Volume Methods
- Spatial and Temporal Discretization
What three important Pre-Processing Concepts?
- Boundary Conditions
- Mesh Generation
- Turbulence Models
What are two important Post Processing Concepts?
- Visualization
- Analysis
What is Method of Lines and what are its benifits?
- Separate discretization in space and time
- Largest flexibility
- Each term can be treated differently to yield different accuracies
What are the three properties of central differencing?
- The variation between centroids is linear
- Second rate accurate
- Unbounded
What is Upwind differenceing?
- Depends on direction of mass flux
- First order accurate
- Bounded
- Diffusive
What is linear upwind differencing?
- We use gradient to improve accuracy of extrapolation
- Second order accurate
- Unbounded
What is the function of a flux limiter?
- To resolve all physical problems as precisely as possible
- To suppress any numerical oscillations
What is total variation diminishing?
Is a measure of how a function changes across grid values
What are two level methods for temporal discretization?
- Forward Euler
- Backward Euler
- Crank Nickelson
What interpolation schemes are used for the convective flux and diffusive flux / pressure gradient?
Linear Upward Differencing / Central Differencing
What kind of interpolation scheme is good for accuracy
/ stability
Linear/Central Differencing
/ Upward Differencing
How do you switch between interpolation schemes?
Blending functions
What problems are caused by Lower / Higher order schemes?
Dissipation / Fluctuation
What does Godunov say?
Linear numerical schemes for solving PDEs having the property of not generating new extrema can be at most first order accurate,
How do flux limiters work?
In non-monotonic regions, switch to upwind differencing, in monotonic regions, retain second order accuracy.
What is total variation diminishing?
Measure of how a function changes across eg. grid values
What are the properties of Minmod?
- Works well for all signals
- Quite dissipative
What are the properties of superbee?
- Generally compressive
- Generates sharp gradients in smooth regions
What are the properties of MUSCL?
- Midpoint between superbee and minmod
What are the properties of two level methods in temporal discretization?
- At most second order accurate
- Only consider values from two different time steps
What are the properties of Forward Euler?
- 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
What are the properties of Backward Euler?
- 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
What are the properties of crank nickelson?
- evaluates at existing and future time level
- Implicit
- 2nd order accurate
What are some higher order explicit methods?
- Adams
- Runge- Kutta
What are some higher order Implicit methods?
LU-SGS
What behavior does the ratio of successive gradients have in monotonic / non-monotonic regions.
Non-Monotonic:
r < 0
Monotonic:
r = 1 for constant gradients
0 < r < 1 for linear gradients
r > 1 non-linear gradients
What are the seven discretization schemes mentioned on the lecture?
- Linear/ Central Differencing
- Upwind Differencing
- Linear Upwind differencing
- Flux limiter schemes
- Flux-vector splitting schemes
- Flux-differencing splitting schemes
- ENO/WENO
What is the CFL condition?
The time step has to be less than or equal to cell size / flow velocity
What is the formula for central differencing?
\phi_f =\psi * \phi_D + (1-\psi) * \phi_C
\psi = (x_f - x_c) / (|x_D - x_C|)
What is the formula for upwind differencing?
F_f = \rho_f (u_f . n) S_f
