Course Syllabus
ATM S 582
Professor Dale Durran
502 ATG Building
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Our goal: Let's try to see the forest and the trees.
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, particularly weather and climate. The class builds on material covered in ATMS 581/AMATH 586, although some familiarity with numerical methods for the solution of differential equations is generally adequate preparation.
We will look closely at behavior of some representative numerical methods in the simplest possible contexts. At the same time, we will strive to develop an overview of the pros and cons of these methods for the simulation atmospheric flows.
Note that each major topic heading (except the first) contains links to important community models in atmospheric science that rely on the method being discussed. We will not cover the actual models in detail, but we will glimpse under the hood at their fundamental numerical methods. These models include the NCAR CESM, ECMWF IFS, NCAR WRF ARW, and GFDL FV3.
Textbook: Durran, D.R., 2010: Numerical Methods for Fluid Dynamics: With Applications in Geophysics. 2nd Ed. Springer-Verlag.
The book is available by chapters for free as pdfs via a subscription through the UW library. That subscription also entitles students to buy $25 paperback copies. To access these priviliges, connect here via a UW computer, and click 'Read Online'.
Lecture Format: Live remote via Zoom MW 9:00-10:20. Lectures will be available as Zoom recordings.
Office Hours: Tues 1:00-2:00 and by appointment (don't hesitate to ask!)
Grading: The grade will be based on a short project and short bi-weekly 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 homeworks. The project will ideally be something related to your research.
PDFs from each lecture are posted here
Course Outline
Basic Strategies for Numerical Approximation to Solutions of PDE
- Grid-point and series-expansion methods (pp. 26-30)
Differential-Difference Equations for the Scalar Transport
- Numerical Dissipation and Numerical Dispersion (pp. 100-110)
- Figures
Further Considerations in Finite Difference Approximations WRF ARW, SAM
- Phase speed as a function of numerical resolution for compact schemes (Fig. 3.10)
- Staggered meshes (pp. 153-157)
- Discrete dispersion relation for systems of equations (pp. 167-169)
- Stability constraints in simulations of multi-dimensional waves (pp. 157-159)
- Splitting into fractional steps (pp. 169-176) Note: we did not cover Strang splitting in class.
- Swirl test
- Effect of Strang splitting (from Skamarock, 2006, MWR, p. 2243)
- Optional background reading on the stability of systems (pp. 148-151)
Finite-Volume Methods GFDL FV3
- Conservation laws and conservation form (pp. 203-206, 211-213)
- Monotone, TVD and monotonicity preserving methods (pp. 213-220)
- Flux-corrected transport (pp. 221-226)
- Flux-limiter methods (pp. 226-235)
- Slope limiters (pp. 238-240)
- Example of how fine scale features are produced in a scalar tracer field by flow deformation
- Postive-Definite Advection Schemes (pp. 271-273)
Spectral and Pseudo-Spectral Models: ECMWF IFS
- Basics of series expansion methods (pp. 281-293)
- Conservation and the Galerkin approximation (pp. 298-299)
- Aliasing error (pp. 178-179)
- Improving efficiency with the transform method (pp. 292-298)
- The pseudo-spectral method (pp. 299-303)
- Spherical Harmonic Functions (pp. 303-308)
- Elimination of the Pole Problem (pp. 308-310)
- Finite-element method, linear elements (pp. 320-323)
- Quadratic elements (pp. 327-336)
The Discontinuous Galerkin and Spectral Element Methods CESM
- h-p methods
- Discontinuous Galerkin method (pp. 341-350)
The Cube Sphere GFDL FV3 , CESM
Semi-Lagrangian Methods ECMWF IFS
- The scalar advection equation (pp. 357-360)
- Backward trajectory
- CFL condition (pp. 98-100)
- 250 hPa winds forecast for March 14, 2015; Todays loop
- Cascade interpolation (pp. 362-365)
- Finite volume integrations with large time steps (pp. 369-372)
- Forcing in the Lagrangian frame (pp. 372--377)
- Comparison with the method of characteristics (pp. 377-379)
Course Summary:
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