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

Class Hours: TR 11:00-12:15 at B131 Van Vleck
Office Hours: TR:2:00-3:00pm, Van Vleck 511, or by appointment
Course Web: www.math.wisc.edu/~jin/Teaching/S06/m837.html

Announcements

Prerequisites: undergraduate numerical analysis and elementary differential equations

References: (reserved in the Library)

• C. Cercignani: The Boltzmann Equation and its Applications, Springer, 1998.
  
• Francois Bouchut, Francois Golse and Mario Pulvirenti: Kinetic Equations and Aymptotic Theory, Gauthier-Villars, 2000.
  
• 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
  
• R.J. LeVeque: Numerical Methods for Conservation Laws, Birkhauser, 1992; Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.
  
• Stanley Osher and Ronald Fedkiw: Dynamic Implicit Surfaces , Springer 2003

Grading: The final grade will be determined by homework assignments including computer projects, and an oral presentation of a research paper assigned by the instructor.

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.

Homework No. 1 , Due March 9, 2006

Homework No. 2 , Due March 23, 2006

Homework No. 3 , Due April 25, 2006

Homework No. 4 , Due

Course materials:

• Physical Problems of different scales:
  * Basic physical equations in Quantum mechanics, classical mechanics, kinetic theory and hydrodynamics
  * The mathematical transition from microscopic to macroscopic scales:
  - The semiclassical limit of quantum mechanics
  - The Grad-Boltzmann limit to derive the Boltzmann equation
  - The fluid dynamical limit of kinetic equation
• 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
  - source terms
  - 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 and fast sweeping methods
  * Applications
  - various applications in geometry and physics
• Numerical Methods for Multiscale Problems
  * Domain decomposition method, heterogeneous multiscale methods, asymptotic-preserving methods, etc.

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