John Trangenstein : Math 225 Scientific Computing II

• a written description of the problem and numerical approach
• a copy of the code
• pictures or numbers for numerical results, and
• a discussion of the results (e.g., describe the accuracy, convergence rate, ...)

• xman. Type "xman&". When the window comes up, click on "Manual page" to see a list of commands, organized by type. To find a particular command, pull down on "Options" to "Search."
• Sun online books. Use a web browser to access http://docs.sun.com/ab2
• Gnu documentation. Type "emacs&". When the window comes up, pull down on "Help" to "Info," then scroll to your favorite topic (e.g., "Make"). Use the middle mouse button to scroll and click.

• Week 1: Gas Dynamics
  • Conservation laws for gas dynamics (Chorin & Marsden, pp1-21)
  • Eulerian and Lagrangian frames of reference
  • Propagating discontinuities
    • Discontinuity surfaces
    • Rankine-Hugoniot jump conditions
    • Jump conditions for gas dynamics
  • Thermodynamics (Courant & Friedrichs, chapter 1)
  • Jump conditions for a polytropic gas
  • Characteristic analysis for gas dynamics
  • Entropy function and flux for gas dynamics
• Week 2: Mathematical theory of hyperbolic conservation laws
  • Riemann problems (Lax monograph)
  • Admissibility conditions for shocks
  • Entropy functions
  • Travelling wave profiles (Smoller, pp 508-)
  • Hyperbolic systems of conservation laws
  • Centered rarefactions
• Week 3: Linear Finite Difference Methods (Leveque)
  • Discrete conservation
  • First-order upwind
  • CFL condition
  • Modified equation analysis
  • Nonlinear conservation laws
  • Lax-Friedrich scheme
  • Godunov scheme
  • Rusanov scheme
• Week 4: Higher-Order Linear Difference Methods
  • Lax-Wendroff process
  • Fourier analysis
  • Norms and convergence
  • Entropy conditions and difference approximations
• Week 5: Nonlinear Schemes
  • Monotonicity-preserving, total-variation-diminishing, monotone
  • Accuracy of schemes at discontinuities
  • Monotonicity-preserving schemes (van Leer)
  • Discrete entropy conditions (LeVeque, p 37f, 133f, 142f)
  • E schemes
  • Flux-corrected transport
  • TVD schemes
  • Slope-limiter methods (MUSCL)
  • LeVeque's large-timestep version of Godunov's method
  • Piecewise-parabolic method
  • Essentially non-oscillatory schemes
• Week 6: Approximate Riemann Solvers
  • Riemann problem for gas dynamics
  • Roe solver
  • Harten-Hyman-Lax entropy fix for the Roe solver
  • Osher-Solomon
  • Bell-Colella-Trangenstein
  • Artificial viscosity
• Week 6: Numerical Methods for Hyperbolic Systems
  • Godunov
  • Rusanov
  • Lax-Friedrichs
  • Lax-Wendroff
  • Fromm
  • Beam-Warming
  • Leap-frog
  • MUSCL
  • Flux limiter methods
  • TVD methods
  • ENO schemes
• Week 7: Multi-Dimensional Methods
  • Operator splitting
  • Unsplit schemes

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Last modified: 5 January 1998