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| Assignments, [reading material], and schedule |
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- 4/2/12: General information; class overview
- 4/4/12: Review method of characteristics.
Introduction to linear vs nonlinear conservation laws. Diffusion
and viscosity solutions.
- 4/6/12: Classical, integral, and weak solutions. Riemann problem.
Lagrangian and Eulerian coordinates.
Worksheet1 due 4/11/12.
[You can use the LaTeX files worksheet1.tex
and amsnumbers.tex .]
Worksheet2 due 4/13/12.
- 4/9/12: Difference between linear and nonlinear conservation laws.
Develop models: Burgers', traffic flow, flood models,
chemical transport and sedimentation, two-phase flow, compressible gas flow.
- 4/11/12: Continue model development.
- 4/13/12: Weak solution to Burgers' equation. Formula for shock speed
for Riemann problem.
Suggested exercise (you can turn it in but it is not required): show
directly that the discontinuous solution with the proper shock speed
is a weak solution.
- 4/16/12: No class today. [Read GLee 12.3]
- 4/18/12: Rarefaction solution to Burgers equation. Other weak
solutions to Burgers equation with Heaviside initial data.
Worksheet3 due 4/27/12.
- 4/20/12: Worksheet in class on construction of solutions to
Burgers' equation with shock overtaking shock or rarefaction catchin
up with shock.
- 4/23/12: Lax-Oleinik solution and examples.
Worksheet4 due 5/4/12.
- 4/25/12: Entropy conditions. Proof of entropy function/flux conditions.
- 4/27/12: Proof of Oleinik chord condition and Lax shock admissibility criterium.
- 4/30/12: Handout ( worksheet6 ) in class: Summary of analysis of
scalar conservation laws in 1D.
Cole-Hopf transformation ( worksheet5 ) for the solution of Burgers equation.
- 5/2/12: Using an Ansatz of travelling wave solution and similarity solution
for Burgers equation leads us to shock and rarefaction
- 5/4/12: Wrap up of scalar conservation laws. Handout:
worksheet6_table. Applications and associated typical behavior of
the solutions to Riemann problem
- 5/7/12: Systems of hyperbolic sonervation laws. Introduction and
linear example. Computation of solutions to Riemann problem by
diagonalization. Animation for the example: hyperbolic_syst.m
- 5/9/12: Characteristics, Hugoniot locus, and intermediate states
for a system and Riemann problem.
- 5/11/12: Examples. Hyperbolic and strictly hyperbolic
problems. Rankine-Hugoniot condition.
Worksheet7 due 5/18/12.
- 5/14/12: Wave equation in 2D-3D. Spherical means.
- 5/16/12: Wave equation in 2D-3D. Spherical means.
- 5/18/12: Hyperbolic systems: examples. Gas dynamics and Equation
of State.
- 5/21/12: Continue examples of nonlinear systems. Shallow water
equations. Linearized systems.
- 5/23/12: Construction of a solution to hyperbolic systems from
shocks, rarefactions, and contact discontinuities. [Examples.]
- 5/25/12: Genuine nonlinear and linearly degenerate fields.
[Examples]. Finish Euler gas dynamics equations.
- 5/30/12: Motion in fluids and solids. Displacement and velocity
gradients. Deformation and stress tensors.
- 6/1/12: [Examples] of fluid motion: computing rate of strain.
- 6/4/12: Derivation of Navier-Stokes equations for Newtonian fluid.
Euler and Stokes equations.
- 6/6/12: Example: flow ina channel as Euler flow, Poiseiile flow,
Darcy flow.
- 6/8/12: Wrap up: asymptotics of Stokes flow, problems with
inetrfaces, and free boundaries (Stefan problem). Review.
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