Lecture Notes

Notes of Interest to the Course

Incompressible flow in elastic wall pipes (This resource may not render correctly in a screen reader.PDF)

Branch points and branch cuts (This resource may not render correctly in a screen reader.PDF)

Conservation laws in continuum modeling (This resource may not render correctly in a screen reader.PDF)

Simplest car following traffic flow model (This resource may not render correctly in a screen reader.PDF)

Discrete to continuum modeling (This resource may not render correctly in a screen reader.PDF)

Weakly nonlinear oscillators (This resource may not render correctly in a screen reader.PDF)

Hopf bifurcations (This resource may not render correctly in a screen reader.PDF)

Weakly nonlinear breathers (PDF)

Stability of numerical schemes for partial differential equations (This resource may not render correctly in a screen reader.PDF)

Lecture Summaries

SES # TOPICS LECTURE SUMMARIES
1

Mechanics of the course.

Example PDE. Initial and boundary value problems.

Well and ill-posed problems.

(PDF)
2

Conservation laws and PDE.

Integral and differential forms.

Closure strategies. Quasi-equillibrium.

(PDF)
3

Classification of PDE.

Examples.

Kinematic waves and characteristics.

(PDF)
4

First order scalar PDE.

Examples of solutions by characteristics.

Domain of influence.

(PDF)
5

Domains of influence and dependence.

Causality and uniqueness. Allowed boundary conditions.

Examples.

(PDF)
6

Graphical interpretation of solution by characteristics.

Conservation. Wave steepening and breaking.

Back to the physics.

(PDF)
7 Region of multiple values. Envelope of characteristics. (PDF)
8

More on envelopes. Infinite slopes at envelope.

Shocks. Conservation and entropy. Irreversibility.

Examples from traffic flow.

(PDF)
9 Continues lecture 8. More examples.
10

Shocks in the presence of source terms. Example.

Riemann problems and Godunov's type methods.

(PDF)
11

The Riemann problem for the kinematic wave equation with convex/concave flux.

Example of a conservation law with a point source term.

(PDF)
12

Shock structure and detailed physics.

Examples: Viscosity solution.

Traffic flow. Flood waves. Shallow water.

(PDF)
13

Shallow water and higher order terms.

Traveling waves, shocks, and the effects of dispersion.

Solitons. Small dispersion limit.

(PDF)
14

PDE and propagation of information.

Equations that allow weak singularities.

Examples.

(PDF)
15

Hyperbolicity and weak singularities.

Examples: Hamilton-Jacobi equation and characteristic form.

Eikonal equation. Multiple values.

(PDF)
16 Continue with Hamilton-Jacobi equation. Characteristics, strips, and Monge cones. Eikonal as characteristic equation for wave equation in 2-D and 3-D. (PDF)
17

Eikonal. Focusing and caustics. Description of the caustic.

Breakdown of approximation. Derivation of amplitude equation.

(PDF)
18

Eikonal. Amplitude and curvature along rays. Behavior near caustic. Caustic expansion.

WKBJ review. Turning points. Conneccion formulas and Airy functions. Matching.

(PDF)
19

First order 1-D systems of equations. Classification.

Hyperbolic systems and characteristics. Domains of dependence and influence. Examples.

(PDF)
20 Examples of first order 1-D hypebolic systems. Linear acoustics. Wave equation. D'Alembert solution. Simple waves. Wave breaking. Shocks and shock conditions. Examples (PDF)
21 Gas dynamics in 1-D. Characteristics, simple waves, Riemann Invariants, rarefaction waves, shocks and shock conditions. Riemann problem. Generalizations to N by N systems. (PDF)
22 Continue with Lecture 21.
23 Linear equations. Superposition. Normal modes and impulse problems (Green's functions). Heat equation in 1-D examples: various initial and boundary value problems. Method of images. (PDF)
24

Green's functions for signaling and source terms.

Heat equation examples. Generalized functions.

(PDF)
25

Generalized functions.

Green's functions for heat equation in multi-D.

(PDF)
26

Green's function. Poisson equation. Stokes equation.

Example: stokes drag on a sphere.

(PDF)