Course Instructor: Shi Jin
Phone: 608-263-3302(o); E-mail: jin@math.wisc.edu

Class Hours: TR 11:00-12:15 at B313 Van Vleck
Office Hours: TR:10:00-11:00am, Van Vleck 511, or by appointment
Course Web: www.math.wisc.edu/~jin/Teaching/S02/m837.html

Announcements

There will be no class on March 19, 21, April 2 and the last teaching week (the week of May 6). We will have make-up classes from 4:30-5:45 in the regular classroom on Tuesdays, Feb 19, 25, March 5, 12, April 9, 16, 23, 30.

Prerequisites: undergraduate numerical analysis and elementary differential equations

References: (reserved in the Library)

• C.A.J. Fletcher: Computational techniques for fluid dynamics , Springer-Verlag, 1991.
  
• E. Godlewski and P.A. Raviart: Numerical approximation of hyperbolic systems of conservation laws , Springer, New York, 1996
  
• J.M. Haile, Molecular dynamics simulation : elementary methods , Wiley, New York, 1992.
  
• R.J. LeVeque: Numerical Methods for Conservation Laws, Birkhauser, 1992
  
• D. C. Rapaport, The art of molecular dynamics simulation , Cambridge University Press, 1995.
  
• J.A. Sethian: Level set methods: evolving interfaces in geometry, fluid mechanics, computer vision, and materials science, Cambridge University Press, 1996

Grading: The final grade will be determined by homework assignments including computer projects.

Homework: Homeworks, both analytic and computational, will be assigned several times during the semester. A basic programing language (Fortran, C, Matlab, etc.) is essential for the computation projects. Students should finish the homeworks with their own effort.

• Homework No. 1 , Due March 12

Homework No. 2 , Due April 4

Homework No. 3 , Due April 25

Homework No. 4 , Due May 14

Syllabus:

• Numerical Methods for Hyperbolic Systems of Conservation Laws and Compressible Flows
  * Mathematical Thoery of Hyperbolic Conservation Laws
  - Linear hyperbolic systems: characteristic method
  - Scalar nonlinear conservation laws: shocks, Rankine-Hugoniot jump condition, entropy condition, Riemann problem
  - Nonlinear systems of hyperbolic equations: riemann problems
  * Numerical Methods for Hyperbolic Conservation Laws
  - stability and convergence theory of finite difference methods for linear problem
  - shock capturing Methods for nonlinear problems: Godunov and Roe methods, kinetic and relaxation methods
  - high resolution shock capturing methods: flux and slope limiters, TVD and ENO methods
  - multidimensional problems: operator splitting
  - Applications: gas dynamics, shallow-water, etc.
• Numerical Methods for Incompressible Flows
  * Introduction to Navier-Stokes equations
  - conservation laws and constitutive relation, scale, incompressible limit
  - various formulations: primitive variable, vorticity and impulse density
  
  * Numerical Methods
  - methods based on primitive varibale formulations: MAC scheme, projection methods, Poisson solver, Gauge method
  - methods based on vorticity formulation
• Level Set Methods for Interface Problems
  * Equations of Motions for Moving Interfaces
  - Theory of front evolution: formulation, total variation, weak solutions and entropy condition, curvature
  - The level set formulation
  * Numerical Approximations to the Leves Set Equations
  - viscosity solutions, numerical schemes for Hamilton-Jacobi equations
  - high resolution schemes for the level set method
  - fast marching method
  * Applications
  - various applications in geometry and physics