Numerical Methods for Conservation Laws
by R. LeVeque

Hyperbolic Systems of Conservation Laws
by E. Godlewski & P.-A. Raviart

Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves
by P. D. Lax

Difference Methods for Initial Value Problems
by R. Richtmyer & K. W. Morton

Advanced Numerical Approximation of Nonlinear Hyperbolic Equations 
Lecture notes by B. Cockburn, C, Johnson, C.-W Shu & E. Tadmor

Tentative Course Plan 

        Week #1

• 04/01  Hyperbolic conservation laws -- an example of traffic flow
• 04/03  Compression and expansion waves
• 04/05  Shock discontinuities and weak solutions

         Week #2

• 04/08  Viscosity regularization and L¹ well-posedness 
• 04/10  Viscosity and entropy, L¹-contraction and monotonicity
• 04/12  Shocks, rarefactions, Riemann problems - the scalar L¹ theory
  

         Week #3

• 04/15 Finite difference methods for transport equations

          Euler schemes -- instability and one-sided differences

          Lax-Friedrichs scheme - numerical viscosity, monotonicity and Godunov theorem

• 04/17   Finite difference schemes cont'd

Lax-Wendroff - the second order numerical viscosity

Leap-Frog - multi-step methods, unitary schemes

Crank-Nicolson - implicit schemes

• 04/19  Stability implies convergence

  The linear theory: CFL condition, amplification symbols, von Neumann stability analysis, numerical dissipation.

  Discontinuous initial data

  Week #4
   

• 04/22 Shock capturing -- the Lax-Wendroff  Theorem

          Consistency, conservation and convergence.

          First generation: numerical viscosity and the class of three-point schemes

• 04/24   Monotone schemes

• 04/26  Beyond monotonicity -- second- and higher-order schemes

          Second generation:  limiters, MUSCL and  TVD schemes with three letters acronym    

Week #5

• 04/29   Systems of conservation laws

          Euler's system, Shallow-water equations and  related examples

          Approaching the limits: Kinetic formulations, Navier-Stokes 

• 05/01   Shocks, rarefactions and the Riemann problem

          Propagation of singularities in linear systems. Nonlinear waves

• 05/03    Viscosity and entropy

          A detour: compensated compactness

          Week #6

• 05/06   TBA
• 05/08   LxF, Godunov and LxW schemes revisited
• 05/10   Godunov type- schemes

          Week #7

• 05/13   High resolution methods

          Tools: limiters, non-oscillatory reconstructions, artificial compression, numerical viscosity

•  05/15  Upwind methods

          Approximate Riemann solvers

• 05/17   Central schemes

          Week #8

• 05/20   Multidimensional extensions

           Dimensional splitting, triangulations, 

• 05/22    Semi-discrete methods

           Runge-Kutta and  multi-step methods, Strong stability

• 05/24    Beyond finite-difference methods

           Finite-volume methods, Finite-element -- streamline diffusion, discontinuous Galerkin, ...

          Week #9

• 05/27    Holiday
• 05/29    Spectral methods

          Fourier and Chebyshev differentiation, filters and Spectral viscosity

• 05/31    Kinetic schemes

          Kinetic formulation. A detour: averaging lemma

          Week #10

• 06/03    Dealing with source terms

          Stiffness and different scales

• 06/05    Incompressible flows

          Side constraints

• 06/07    Review 

          Final Exam

• 06/11  3-6pm