ch2-partB summery Flashcards
Space difference schemes are used to:
- March forward in space
Time difference schemes are used to:
- March forward in time
Time difference schemes meet:
- The accuracy requirement at the first and second order
- Higher order schemes appear cumbersome and are not widely used in NWP
Temporal differencing schemes:
- Explicit
- Model prognostic equations are approximated using finite differences so that the variables at the future time appear only on one side of the equation
- Implicit
- Variables at the future time appear on both sides of the equation
Leapfrog time differencing:
- An example of explicit schemes
To obtain leapfrog time differencing equation:
- Apply centered finite difference approximation in time and space to partial derivatives
To obtain implicit equations:
- Apply:
- Backward finite difference in time and
- Centered finite difference in space
The resulted equation (from implicit scheme):
- Implicit time differencing scheme
Courant number:
- Limiting value that is necessary for the time differencing schemes to produce numerically-stable solutions
- Non-dimensional
CFL conditions:
- Limiting value of the courant number
CFL conditions represent:
- The maximum value of the courant number that permits numerically-stable model solutions
The exact value of the CFL conditions:
- Vary
The exact value of the CFL conditions varies depending on:
- Spatial and temporal finite different scheme utilized
Choosing fine grid spacing result on
- Small steps between intermediate forecasts leading to an increase in
- The number of time steps and
- Computing time
According to CFL conditions fine mesh model forecasts require:
- More number of intermediate time steps and therefore
- Need powerful computers
Approximating 1D moisture advection equations in FD form can be done using:
- Forward in time and
- Centered in space (FTCS) formulas
If the sign behind the advection term is negative and if it is positive:
If the sign is negative (from low to high)
If the sign is positive (from high to low)
What is leapfrog time differencing scheme? How does it work?
- An example of explicit schemes
- Obtained when you apply centered finite difference approximation in time and space to partial derivative
- Usually used when you don’t have past values
How to limit resolution:
- If /_\ x is small
- /_\ t is small
- Number of time steps is large
- Fine resolution
- Number of time steps is large
- /_\ t is small
If u=t:
- The wind moves from left to right
- Values are decreasing
