18.311 | Spring 2014 | Undergraduate

Principles of Applied Mathematics


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

Course Info

As Taught In
Spring 2014
Learning Resource Types
Problem Sets
Lecture Notes