Despite confusing title, this class is
not about how to simulate atmosphere or ocean flows using models.
Rather, it is an introduction to numerical methods of solving
differential and general nonlinear equations, which are common to many
geophysical problems, as well as to advanced data analysis methods.
Instructors
Prof. Marat Khairoutdinov
Office hours
Endeavour 121; by email appt
Email
marat.khairoutdinov@stonybrook.edu
Textbooks
The course does not strictly follow any textbook. In fact, the field of
computational fluid dynamics and related techniques is too broad to be
covered in a single textbook. Therefore, it is important to take class
notes. However, here is a short list of recommended books:
Fletcher, C. A. J., 1991: Computational Techniques for Fluid Dynamics,
Vol. 1. 2nd ed. Springer-Verlag, 401 pp.
Durran, D. R., 1999: Numerical methods for wave equations in
geophysical fluid dynamics. Springer, 465 pp.
Daley, R. 1991: Atmospheric data analysis. Cambridge
University Press, 455pp
Grading
50% for exams (2) and 1 midterm project, 50% for homework
assignments. Final grading will be based on the average of the
three section scores.
Special Note
Hands-on experience is the best way to learn numerical methods.
Homework will involve writing simple programs and plotting the results.
You will need to have access to computers with programming and graphing
software. Knowledge of high-level compiled (e.g., Fortran (preferred),
C) or scripting (e.g., IDL, Matlab) computer languages is required for
this course. It is, however, up to you which programming language or
graphing application to use.
Outline
Fundamentals of Finite-Difference Schemes
Definitions of consistence, convergence, and stability; First and
second order derivatives; Construction of higher order approximations;
Numerical solution of nonlinear equations.
Methods for Initial-Value Problems of Linear Partial Differential
Equations
Linear computational stability analysis; Classification and canonical
forms; Basic numerical schemes for advection and diffusion equations;
Upstream and downstream biased schemes; Time-integration schemes;
Time-splitting and directional splitting schemes; Implicit and explicit
schemes; Numerical diffusion and dispersion; Extension to multiple
dimensions; Grid systems.
Methods for Nonlinear Initial-Value Problems
Fourier representation of discrete fields; Nonlinear interaction and
instability; Methods to eliminate nonlinear instability; Construction
of conservation schemes; Monotonic and positive definite schemes;
Barotropic vorticity model; the Arakawa Jacobian; Basic concepts of
spectral methods; Semi-Lagrangian and finite-volume methods
Methods to Solve Elliptic Equations
Fourier method; Relaxation methods; Multi-grid methods; Tri-diagonal
matrix solver
Data Analysis
Classical objective analysis; Statistical estimation; Maximum
likelihood estimation;
Least variance estimation; Kalman filtering; Statistical spatial
interpolation; Variational analysis method; Adjoint models;
Multivariant analysis
Homework
Presentations
Coding Efficiency
Americans
with Disabilities Act
If you have a physical, psychological, medical or learning disability
that may impact your course work, please contact Disability Support
Services, ECC (Educational Communications Center) Building, room 128,
(631) 632-6748. They will determine with you what accommodations are
necessary and appropriate. All information and documentation is
confidential.
Students requiring emergency evacuation are encouraged to discuss their
needs with their professors and Disability Support Services. For
procedures and information, go to the following web site.
http://www.ehs.sunysb.edu/fire/disabilities/asp
Academic Integrity Statement
Each student must pursue his or her academic goals honestly and be
personally accountable for all submitted work. Representing another
person's work as your own is always wrong. Any suspected instance of
academic dishonesty will be reported to the Academic Judiciary. For
more comprehensive information on academic integrity, including
categories of academic dishonesty, please refer to the academic
judiciary website at http://www.stonybrook.edu/uaa/academicjudiciary/
Adopted by the Undergraduate Council
September 12, 2006