Advanced Numerical Modeling of Geophysical Flows
MWF 11:30-12:20, Room ATG 610
Textbook:
Durran, D.R., 1999: Numerical Methods for Wave Equations in
Geophysical Fluid Dynamics. Springer-Verlag.
Overview:
The purpose of the course is to obtain a deeper understanding of the
basic numerical techniques that form the foundation for the computer
models commonly used to simulate geophysical flows. This course is
taught every-other year and builds on material covered in ATMS
581/AMATH 586.
Homework
Exercise 1: Due Wednesday, Feb 7th.
Exercise 2: Due Wednesday, Feb 21st.
Exercise 3: Due Friday, March 9th
Course Outline
Review: Differential-Difference Equations for the Scalar Wave Equation
- Numerical Dissipation and Numerical Dispersion (pp. 72-80)
- Dispersion of a spike (mpeg)
- 1st, 2nd, 4th order spike (mpeg)
- 1st, 2nd, 4th order sum of 7.5 and 10 Dx waves (mpeg)
- 1st, 2nd, 3rd order spike (mpeg)
- 1st, 2nd, 3rd order sum of 7.5 and 10 Dx waves (mpeg)
Further Considerations in Finite Difference Approximations
- Example of how fine scale features are produced in a scalar tracer field by flow deformation
- Staggered meshes (pp. 113-117)
- Stability constraints in simulations of multi-dimensional waves (pp. 117-119)
- Discrete dispersion relation for systems of equations (pp. 126-128)
- Splitting into fractional steps (pp. 129-136)
Finte-Volume Methods
- Conservation laws and conservation form (pp. 241-244, 249-250)
- Monotone, TVD and monotonicity preserving methods (pp. 251-257)
- Flux-corrected transport (pp. 257-263)
- Flux-limiter methods (pp. 263-271)
- ENO and WENO methods (notes)
- Comparisons of Lax-Wendroff based flux-limiter and ENO solutions for advection
Spectral Models on
the Sphere
- Basics of series expansion methods (pp.173-176)
- Spherical Harmonic Functions (pp. 195-200)
- Elimination of the Pole Problem (pp. 200-202)
Approximating Open
(Wave-Permeable) Boundaries
- The 1D shallow-water system and well-posed boundary conditions (pp. 295-400)
- The radiation condition (pp. 400-402)
- Spurious reflections and stability conditions for numerical approximations to initial-boundary-value problems (pp. 402-412)
- Extensions
to two-dimensional shallow-water flow (pp. 412-419)
- Extensions to two-dimensional stratified flow (pp. 419-431)
- Wave absorbing layers (pp. 431-437)
Semi-Lagrangian Methods
- The
scalar advection equation
- Forcing
the Lagrangian frame
- Comparison
with the method of characteristics
- Flux-form representation of tracer advection
Grading: The grade will be based on a short project and three homework assignments, one of which will be identified as a take-home midterm that must be done independently. You may work with other students on the other two homeworks. The project will ideally be something related to your research.