E4300: CM1 --
        Numerical Methods

Time:

Tu-Th 11:00-12:15 Location: 327 Mudd
Instructor: Marc Spiegelman   (home page)
Office Hours: Tues/Thurs 1:45-2:45pm
 211 Mudd (APAM)
TAs: Mike Jenkinson
Office Hours: Mon/Wed 11:00-12:00
292 Mudd

Texts:
Principal Texts:
Online References/Resources:
Other Dead Trees
Grading: Homework 60%, Projects 40%

All Homeworks due by 5pm on the due date in the E4301 Homework box in 200 Mudd (APAM).
Homeworks can also be turned in during class.
Matlab Resources

Other Software Resources


Prerequisites:

This course expects  fluency in the basic applied math required for Numerical Methods 4300 (i.e. linear algebra, calculus, ODE's and some programming experience) plus a course in PDE's to the level of E3102. 

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Syllabus
Dates Subject Supplemental Material
6 Sep
  • Introduction to Numerical PDE's: Logistics, Objectives,
    • The Big Picture = Intelligent Choices
8-13 Sep
  • Introduction to Modeling:  Conservation of Anything
  • Some Example Model systems
15-22 Sep
  • Review of Basic Numerical Methods for ODE Initial Value problems
    • Basic stepping algorithms and Error analysis
    • Stiff Systems, Matlab ODE routines
27 Sep -
  6Oct

  • Hyperbolic Equations in 1-D: Wave Equations
    • Analytic solutions, wave behavior and method of characteristics
    • Basic Eulerian methods, FD, FV -- error and stability analysis
    • Advanced Algorithms: Semi-Lagrangian methods, Pseudo-Spectral, TV and flux limiters
11-18 Oct
  • Parabolic Equations in 1-D: Diffusion Equations
    • Analytic behavior, Basic FD,FV algorithms -- error stability analysis
    • Explicit vs Implicit solvers, Crank-Nicolson schemes
    • Introduction to Finite Elements and Spectral methods, Method of lines
20-27 Oct
  • Introduction to Multi-Physics problems -- advection, diffusion, reaction in 1-D
    • Basic algorithms and issues
    • Advanced combined algorithms, SLCN, Non-linear problems
  • Old MMM Notes for multi-physics problems
  • Leveque Ch 11
  • Spiegelman & Katz, paper on SLCN
  • Knoll & Chacon paper on multi-physics and splitting problems
  • 1-d SLCN  Demos
  • A non-linear Diffusion/Radiation problem
  • Problem Set #5 (due Nov 15)
    • Choose 1 problem (pdf)
1 Nov
Election Day (no Class)...go vote!

3-15 Nov
  • Higher Spatial Dimensions
    • Elliptic Boundary Value problems: FD, FV, FEM methods -
    • Parabolic Eqations in 2 and 3-D, explit vs implict methods
    • Semi-Lagrangian Methods FD vs FEM
18 Nov -
  1Dec
  • Numerical Linear Algebra for large Sparse Linear/Non-Linear systems
    • Direct methods - Sparse Direct
    • Basic Iterative methods -  analysis, convergence
    • Preconditioned Krylov methods - CG, GMRES
    • MultiGrid Methods
    • Advanced Software - PETSc
  • Leveque Ch 4
  • Trefethen and Bau
  • Jonathan Shewchucks CG notes
  • Briggs: A multi-grid tutorial
  • Old MMM Notes for Multi-grid
  • Lecture PDF
  • Petsc Poisson3D Examples Directory
  • Problem Set #6 (Extra Credit)
    • Problem 1 required  (pdf)
    • Blankenbach et al Benchmark paper (pdf)
6-8 Dec
  • Putting it all together: Multi-physics problems & Non-linear problems
    • Advanced software for multi-physics problems
  • A convection movie (Hybrid FEniCS/PETSc code with Fieldsplit Block Preconditioners)
16 Dec
Projects Due


marc spiegelman
Last modified: 3 Sept 2010