Syllabus

Course Meeting Times

Lectures: 3x / week, 1 hour / session

Prerequisites

18.06 Linear Algebra

Description

This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat/diffusion, wave, and Poisson equations.

Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems including operator adjoints and eigenproblems, series solutions, Green's functions, and separation of variables.

Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems, including stability and convergence analysis and implicit/explicit time-stepping.

Textbook

There is no required text for this course, though the following book is recommended (emphasized more the numerical part of the course).

Amazon logo Strang, Gilbert. Computational Science and Engineering. Wellesley, MA: Wellesley-Cambridge Press, 2007. ISBN: 9780961408817. More information, including on-line chapters, can be found on Prof. Strang's CSE Web site.

Assignments

Late problem sets are not accepted, however the lowest problem set score will be dropped.

Grading

ACTIVITIES PERCENTAGES
Assignments 50%
Midterm exam 20%
Final exam 30%

Calendar

SES # TOPICS KEY DATES
L1 Overview of linear PDEs and analogies with matrix algebra Problem set 1 out
L2 Poisson's equation and eigenfunctions in 1d: Fourier sine series  
L3 Finite-difference methods and accuracy  
L4 Discrete vs. continuous Laplacians: symmetry and dot products

Problem set 1 due

Problem set 2 out

L5 Hilbert spaces and adjoints for differential operators  
L6 Algebraic properties of the 1d Laplacian: consequences for Poisson, heat, and wave equations  
L7 Laplacians in higher dimensions, and general Dirichlet and Neumann boundary conditions

Problem set 2 due

Problem set 3 out

L8 Separation of variables, in time and space  
L9 Separation of variables in cylindrical geometries: Bessel functions  
L10 Multidimensional finite differences and Kronecker products

Problem set 3 due

Problem set 4 out

L11 Rayleigh quotients, the min-max theorem, and estimating the first few eigenfunctions  
L12 Green's functions and inverse operators  
L13

Green's function of the 1d Laplacian

Reciprocity

Problem set 4 due

Problem set 5 out

L14 Delta functions and distributions I  
L15

Delta functions and distributions II

Green's functions via delta functions

 
L16 Green's function of the 3d Laplacian

Problem set 5 due

Problem set 6 out

L17 The method of images, interfaces, and surface integral equations  
L18 Green's functions in inhomogeneous media: Integral equations and Born approximations

Problem set 6 due

Problem set 7 out

L19

Dipole sources and approximations

Overview of time-dependent problems

 
L20 Time-stepping and stability: Definitions, Lax equivalence  
L21 Von Neumann analysis and the heat equation  
L22

Explicit and implicit timestepping, and Crank-Nicolson schemes

Wave equations in first-order form

 
L23

Algebraic properties of wave equations and unitary time evolution

Conservation of energy in a stretched string

 
Midterm exam
L24

Wave equations in higher dimensions

D'Alembert's solution and planewaves

Problem set 7 due

Problem set 8 out

L25 Staggered discretizations of wave equations  
L26

Wave propagation examples

Phase and group velocity via Fourier analysis

Problem set 8 due

Problem set 9 out

L27

Group velocity dispersion

Waveguides with hard walls

 
L28 Reflection and refraction, evanescent waves, and dispersion relations

Problem set 9 due

Problem set 10 out

L29 Waveguide eigenproblems  
L30 Maxwell's equations  
L31 Numerical simulation of Maxwell's equations: computational electromagnetism  
L32 Wave equations in frequency domain: Helmoltz equations and Green's functions

Problem set 10 due

L33 Perfectly matched layers (PML)  
L34

PML in the time domain

Finite element methods: introduction

 
L35 Galerkin methods  
L36 Tent functions and recovering finite-difference methods from the Galerkin approach  
L37 Symmetry and linear PDEs  
Final exam