Principles of Applied Mathematics

Calendar

LEC #	TOPICS	KEY DATES
1	Overview of Class
2	General Framework. Discrete to Continuum Limits.
3	Sources for Conservation Laws
4	Conservation Laws in 2-D or 3-D, Dimensional Analysis, Fick’s Law	Problem Set 1 Out Next Day
5	Conservation Laws and Partial Differential Equations (PDEs), Method of Characteristics, Traffic Flow
6	Linearized Traffic Flow
7	Characteristic Curves, Nonlinear Problems
8	Nonlinear Kinematic Waves and Wave-Breaking
9	Traffic Flow Examples	Problem Set 1 Due, Problem Set 2 Out
10	Shock Jump Condition
11	Domain of Dependence, Characteristics, Shocks
12	Flood Waves	Problem Set 2 Due Next Day, Problem Set 3 Out
13	Ordinary Differential Equations (ODEs), Shock and Rarefaction
14	Arrow of Time
15	Extended Theory
16	EIKO, Waves and Caustics
17	Whole Curves, Envelopes, Mach Cone	Problem Set 3 Due
18	Domain of Dependence, 1st Order PDEs	Problem Set 4 Out Day Before
19	Review of Theory: Expansion Fans, Shocks
20	Review of Theory: Entropy Increase at Shocks
21	Entropy, Quadratic Flow
22	Envelopes and Relationship Between Conservation Laws and Shocks
23	Shock Structure Due to Higher Order Effects	Problem Set 4 Due, Problem Set 5 Out Next Day
24	Gas Dynamics, Acoustics, and Strings
25	Gas Dynamics, Polytropic Gas, Hyperbolic
26	Linear Algebra for Gas Dynamics
27	Linear Gas Dynamics: Acoustic in a Pipe	Problem Set 5 Due Next Day
28	Gauss / Stokes Theorem	Problem Set 6 Out
29	General Solution for the Wave Equation
30	Multivariable Calculus, Differentiability, Properties of the Gradient, Taylor Expansion, Implicit Function Theorem
31	Characteristics for Klein-Gordon	Problem Set 6 Due, Problem Set 7 Out Next Day
32	Characteristics, Simple Waves, and Shocks for Gas Dynamics, Numerical Method by Characteristics
33	Separation of Variables and Normal Modes
34	Scalar Product Definition of Self-Adjoint
35	Generalize to Operators, Finite Differences Schemes for PDEs	Problem Set 7 Due Next Day, Problem Set 8 Out Next Day
36	FD Schemes
37	Introduction to the vNSA Problem Series
38	Artificial Viscosity
39	von Neumann Stability Analysis for Numerical Schemes	Problem Set 8 Due