Finite-difference Errors - Part 3: Phase & Group-speed Errors, Nonlinear stability

Under construction (this web page, and most other web pages for this course).

Instructor: Roland Stull

Learning Goals:

• (finish any learning goals from last time)
• define numerical dispersion and its effects in NWP.
• explain what the parasitic (computational / ghost) mode is, and how it affects NWP.
• explain how erroneous group velocity affects NWP.
• design an NWP system (domain size, grid spacing, finite-difference scheme, computational resources needed, time required
• explain the utility of adaptive time steps in WRF.
• discuss nonlinear error issues such as aliasing, and the associated accumulation of energy at small wavelengths.
• explain the pros and cons of using a "positive-definite" advection scheme.
• use the Piecewise Parabolic Method (PPM) for advection, and explain its pros and cons.

Readings BEFORE class:

1. See handout on Nonlinear Errors in NWP.
2. Skim the journal paper: Colella, P., and , P. R. Woodward, 1984: The piecewise parabolic method (PPM) for gas- dynamical simulations. J. Comp. Phys., 54, 174-201.

Homework AFTER class:

1. Do Homework #6 (see details at this link). Advection of a 1-D pollutant puff.
   

Topics

C. Amplitude Errors (continued)

   (Catch up for any remaining topics from last week.)

D. Phase- & Group-speed Errors

1. Phase speed errors: forward-in-time backward-in-space.
2. Computational (parasitic) modes for  3-point centered in time and centered in space.
3. Group speed errors
4. Adaptive time step in WRF.

E. Nonlinear Errors

1. DFT
2. Aliasing
3. Nonlinear physics such as advection
4. Energy spectrum
5. Nonlinear Instability

D. Piecewise Parabolic Method (PPM) for Advection