Instructor:   Dr. Brian Jewett Office:   202 Atmos. Sci. Bldg., 333-3957 E-mail:   bjewett@illinois.edu Days/Time:   9:30-10:50am Tu/Th Location:   109 Atmospheric Sciences Bldg Credit:   4 hours Prerequisites:   Math 380 or consent of instructor   click for PDF handout (JPG, PNG)

Right: Visualization of the final course problem from last year (demo configuration).

click for full-sized movie   Shown on left: (1) surface Y-component winds (shaded, and vectors); (2) potential temperature surface showing colliding density currents; (3) vertical velocity (red +, blue -) mapped onto ~0.1/sec vertical vorticity surface. On Right: closeup of early evolution of vorticity sheet, seen from nearly above; ground plane shows surface winds, and cross section is of perturbation temperature. There are +Y winds (orange) in the right half of the domain, and -Y (blue) in the left. A vortex sheet develops between colliding density currents. Small perturbations grow and vorticity is stretched as convergence and vertical motion concentrate near the center axis. Merging of nearby vortices results in upscale growth to fewer, larger rotation centers before the solution decays.

FOR: This course is for those interested in numerically solving partial differential equations that describe compressible fluid flow, utilizing high performance computers at the National Center for Supercomputing Applications (NCSA) at the University of Illinois.

Key objectives: that those taking the course leave it with -

1. A thorough understanding of the fundamentals - the basis for choosing and evaluating numerical methods
2. The ability to critically interpret numerical methods as presented in the literature. We will work through several papers and examine their descriptions of their methods and how they are assessed in terms of stability, accuracy, and error characteristics
3. Most important: the ability to apply these methods to high-performance computers. There will be no "black boxes" (other than visualization packages) in the course - the emphasis is on coding and understanding numerical method behavior as applied to linear and nonlinear fluid flow problems in 1-D, 2-D and 3-D settings.

COURSE OVERVIEW: The course focuses on the use of numerical methods in solving wave equations. Content is directed at understanding how finite difference, finite volume and semi-Lagrangian methods affect the solution of advection and Burger's equations. Topics include time and space approximations, use of staggered meshes, nested grid implementation and limitations, temporal and directional splitting, monotonicity, positive definiteness, and flux limiting. Nonlinear systems including the shallow-water, Euler equations, and quasi-compressible systems are discussed. Throughout the course, findamental principles such as stability, accuracy, convergence, nonlinear instability and aliasing are introduced and are related to the behavior of different numerical approximations.

COMPUTER PROBLEMS: High speed computers will be used to solve fluid flow problems in one, two and three dimensions, using regular and nested grid approaches. We will emphasize writing clear and effective programs, as well as structuring codes for efficient use of parallel computers. Course assignments may be programmed in Fortran or C, and introductory codes and plotting programs in both languages will be provided. The behavior of the numerical solutions will be compared to known solutions when they are available.

TEXT: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, by Dale Durran, Springer-Verlag New York, Inc., 1999. (Required)

INTRO: Welcome; I am Dr. Brian Jewett. I teach and carry out research in the Atmospheric Sciences Dept. My specialty is 3d numerical modeling of a variety of phenomena - severe thunderstorms and squall lines, hurricanes, and heavy snowstorms. In addition, this winter I'll be forecasting for and flying on the NCAR C-130 for observations of Midwest cyclones.

If you are considering taking 502/505/566, read on:

• You need an understanding of calculus, some matrix algebra, and preferably exposure to PDEs.
• You should be comfortable with a programming language, or willing to learn. This class could be abrupt if you have no programming experience at all, as we get going fairly quickly. To help everyone get started and to begin at a common starting point, I will pass out an introduction (sample) program at the start of class (in Fortran and also in C) which will serve as a basis upon which you will build your later programs.
• The computing objectives are (a) getting everyone comfortable and familiar with our programming environment on a machine at NCSA, (b) getting started with 1-D codes before we add complexity, and (c) working up to 3-D nonhydrostatic nonlinear problems by the end of class. Each class computer problem will be designed to build on the last to make understanding and completing the assignments more straightforward for all.
• There will be a number of homework assignments (for which I encourage students to work together - but turn in your own work), and three exams (2+final) during the semester.
• I am investigating use of fairly sophisticated visualization tools. I will introduce these if I feel there is time, if the tools are practical, useful and reasonably straightforward to learn and apply, and provided they fit into the class objectives.

If you have any questions about the class, please feel free to contact me.