Math 226 Numerical Methods for Partial Differential Equations

Numerical Solution of Hyperbolic Partial Differential Equations, Cambridge U Press September 2009(Local users only)
Numerical Solution of Elliptic and Parabolic Partial Differential Equations(Local users only)

John Trangenstein : Math 226 Numerical Methods for Partial Differential Equations

Class hours: MWF 10:20-11:10 Physics 205
Office hours: MWF 1:00-2:00 Physics 029C

To get online help with commands and utilities, here are some useful sources:

xman. Type "xman&". When the window comes up, click on "Manual page" to see a list of commands, organized by type. To find a particular command, pull down on "Options" to "Search."
Gnu documentation. Type "emacs&". When the window comes up, pull down on "Help" to "Info," then scroll to your favorite topic (e.g., "Make"). Use the middle mouse button to scroll and click. You will have to create a .emacs file in your home directory to make this work properly (you can copy ~johnt/.emacs):
- pushd /usr/local/egcs/info
- info
- mGdb (to see gdb documentation)
To see g77 documentation, replace "mGdb" with "mg77". To see egcs documentation, replace "mGdb" with "megcs".
Linux online books.
Netlib Home Directory

News flashes:

To get a math department computer account, go to Departmental Computing
To get copies of the deal library source code:
- edit your ~/.cshrc file and add the following lines at the end:
  setenv CVSREAD
  setenv CVSROOT /home/faculty/johnt/cvs
- make a directory for this course, and cd to that directory.
- type
  cvs get memdebug graphics gui gnu parabolic
  cvs get deal
  cp ~johnt/math226/GNUmakefile .
  make
- to run the first heat equation solver, type
  1d/heat input
  This will open a window on your screen (hit ENTER to remove it) and write some .xpm files in your directory.
- to make a movie of the run, type
  convert *.xpm movie.gif
  animate movie.gif
- Your movie might look like the following.
  Warning: the .gif files can be pretty big. You will have to watch your disk usage.
  $ANIMATION NOT VIEWABLE OUTSIDE DUKE$
- you can save the graphics in a window by typing
  eps4paper results
  This will create the file results.eps, which you can include in a LaTex file and massage easily. If you want to print the graphics by itself, try
  convert results.eps results.ps
  lpr results.ps
- you can play with the input parameters by editing the file named input. In particular, you might play around with values for ncells and cfl.
To enable your sessions to dump cores, add
unlimit coredumpsize
to your .cshrc file.
To see your default printer and its location, type
lpq
Some useful printers in the math department are
- lw0 in room 136
- lw3 in room 250
To print a window created by the class program, run xv:
- click in xv 3.10a(PNG)... with the right mouse button
- click on Grab in xv controls with the left mouse button
- change the delay to 1 sec in the xv grab window
- click on Grab in xv grab with the left mouse button
- after the beep, click in the linear advection window with the left mouse button
- click on Print in xv controls with the left mouse button
- click on B/W in xv prompt with the left mouse button
- click on OK in xv postscript with the left mouse button
You will have to work a bit to get GNU documentation to work properly. See the online help discussion below.

How to turn in computer programs:

Make a directory called ch2 for the homework in chapter 2 and put your code there. Use a different directory for each chapter.
chmod a+rx parabolic
chmod a+r all source code in ch2.
chmod a+x all executables in ch2.
Put all written assignments on paper and turn them in.
Include a printed copy of your program, the name of the executable, and a sample output.
Put all computational output on a web page and tell me its location.
Send your program by email to johnt@math.duke.edu
It would be best if you put only your homework in ch2; put test code elsewhere.

Course Synopsis:

Scalar Hyperbolic Conservation Laws
- Linear Advection
  - Conservation Law on an Unbounded Domain 8/24
  - Integral Form of the Conservation Law 8/24
  - Advection-Diffusion Equation 8/24
  - Advection Equation on a Half-Line
  - Advection Equation on a Finite Interval
- Linear Finite Difference Methods
  - Basics of Discretization 8/26
  - Explicit upwind differences 8/26
  - Explicit Downwind Differences 8/28
  - Implicit Downwind Differences 8/28
  - Implicit Upwind Differences 8/28
  - Explicit Centered Differences 8/28
- Modified Equation Analysis
  - Explicit Upwind Differences 8/28
  - Explicit Downwind Differences 8/28
  - Explicit Centered Differences 8/28
- Consistency, Stability and Convergence
- Fourier Analysis
  - Constant Coefficient Equations and Waves 8/28
  - Dimensionless Groups 8/28
  - Finite Differences and Advection 9/2
  - Fourier Analysis of Individual Schemes 9/2
- L2 Stability for Linear Schemes
- Lax Equivalence Theorem 9/4
- Measuring Accuracy and Efficiency 9/4
Nonlinear Scalar Laws
- Nonlinear Hyperbolic Conservation Laws
  - Nonlinear Equations on Unbounded Domains 9/7
  - Characteristics 9/7
  - Development of Singularities 9/7
  - Propagation of Discontinuities 9/7
  - Traveling Wave Profiles 9/7
  - Entropy Functions 9/7
  - Oleinik Chord Condition 9/9
  - Riemann Problems 9/9
  - Galilean Coordinate Transformations 9/9
- Case Studies
  - Traffic Flow
  - Miscible Displacement
  - Buckley-Leverett Model
- First-Order Finite Difference Methods
  - Explicit Upwind Differences 9/9
  - Lax-Friedrichs Scheme 9/9
  - Timestep Selection
  - Godunov's Scheme
  - Comparison of Lax-Friedrichs, Godunov and Rusanov
- Nonreflecting Boundary Conditions
- Lax-Wendroff Process
- Other Second-Order Schemes
Nonlinear Hyperbolic Systems
- Theory of Nonlinear Hyperbolic Systems
  - Characteristics 9/14
  - Propagating Discontinuity Surfaces 9/14
  - Frames of Reference 9/14
  - Change of Frame of Reference for Propagating Discontinuities 9/14
  - Lax Admissibility Condition 9/16
  - Hugoniot Loci 9/16
  - Centered Rarefactions 9/16
  - Riemann Problems 9/16
- Upwind Schemes for Gas Dynamics
  - Lax-Friedrichs 9/18
  - Rusanov 9/18
  - Godunov 9/18
- Parabolic Equations
  - Theory
    - Continuous Dependence on the Data 9/21
    - Green's Function 9/21
    - Reflection and Superposition 9/21
    - Maximum Principle 9/21
    - Bounded Domains and Eigenfunction Expansions 9/21
  - Finite Differences
    - Continuous-In-Time Methods 9/21
    - Explicit Centered Differences 9/23
      - Stability
      - Explicit Finite Element Scheme
      - Consistency
      - Convergence
      - Finite Difference Programs
    - Implicit Centered Differences 9/25
    - Crank Nicolson Scheme 9/25
    - Classical Higher-Order Temporal Discretization 9/25
    - Deferred Correction 9/28
  - Lax Convergence Theorem
  - Fourier Analysis 9/28
    - Symbols
    - Consistency
    - Stability
    - Interpolation and Truncation
    - Convergence Theorems
  - Measuring Accuracy and Efficiency
  - Finite Difference Methods in Multiple Dimensions 9/28
    - Unsplit Methods
    - Operator Splitting
- Iterative Linear Algebra
  - Relative Efficiency of Implicit Computations 9/30
  - Matrix Norms 9/30
  - Neumann Series 9/30
  - Perron-Frobenius Theorem 10/2
  - M-Matrices 10/2
  - Iterative Improvement
    - Richardson's Iteration 10/7
    - Jacobi Iteration 10/7
    - Gauss-Seidel Iteration 10/7
    - Successive Over-Relaxation 10/9
    - Termination Criteria for Iterative Methods 10/9
  - Gradient Methods
    - Steepest Descent 10/9
    - Conjugate Gradients 10/12
    - Preconditioned Conjugate Gradients 10/12
  - Minimum Residual Methods
    - Orthomin 10/12
    - GMRES 10/14
  - Nonlinear Systems
    - Newton Algorithms 10/16
    - Nonlinear Krylov Algorithms 10/16
    - Nonlinear Case Studies 10/16
  - Multigrid
    - Multigrid Algorithm 10/19
    - Coarse Grid Projection 10/19
    - W-Cycle 10/19
    - Work Estimates 10/19
    - Condition Number 10/19
    - Prolongation 10/21
    - Multigrid Debugging Techniques 10/23
- Introduction to Finite Elements
  - Weak Formulation 10/26
  - Applications 10/26
    - Steady-State Heat Flow
    - Incompressible Single-Phase Flow in Porous Media
    - Linear Elasticity
    - Hyperelasticity
    - Electro-Magnetism
  - Galerkin Methods 10/26
  - Finite Element Example 10/28
    - Nodal Viewpoint
    - Element Viewpoint
    - Finite Differences
  - Overview of Finite Elements 10/30
  - Reference Shapes 11/2
    - Intervals
    - Triangles
    - Quadrilaterals
    - Tetrahedra
    - Wedges
    - Hexahedra
  - Polynomial Families 11/2
    - Lagrange Polynomials
    - Legendre Polynomials
    - Hierarchical Polynomials
  - Shape Function Familes 11/4
    - Lagrange Shape Functions
    - Hierarchical Shape Functions
      - Intervals, Squares or Cubes
      - Barycentric Monomials
      - Triangles
      - Tetrahedra
      - Wedges
      - Shape Function Programs
  - Quadrature Rules 11/6
    - Newton-Cotes Quadrature
    - Gaussian Quadrature
    - Lobatto Quadrature
    - Tensor Product Quadrature
    - Integrals in Barycentric Coordinates
    - Quadratures on Triangles
    - Quadruatures on Tetrahedra
  - Mesh Generation
  - Coordinate Mappings 11/9
    - Triangles
    - Quadrilaterals
    - Hexahedra
    - Tetrahedra
    - Wedges
    - Continuity
  - Finite Elements 11/11
  - Finite Element Linear Systems 11/11
    - Integrals for the Differential Equation Inhomogeneity
    - Integrals for the Differential Operator
    - Neumann Boundary Conditions
    - Essential Boundary Conditions
    - Linear System Assembly
  - Finite Element Program Examples 11/13
- Finite Element Theory
  - Norms and Derivatives
    - Vector and Function Norms
    - Function Spaces
    - Differentiation
  - Sobolev Spaces
    - Multi-Indices
    - Sobolev Norms
    - Imbedding Theorems
    - Hilbert Scales
    - Extension Theorem
    - Trace Theorems
    - Poincare Inequality
    - Friedrichs Inequality
  - Elliptic Equations
    - Elliptic Differential Operators
    - Green's Formula
    - Dirichlet Problems
  - Elliptic Regularity
    - Coercivity
    - Well-Posedness
    - Gaarding's Inequality
    - Higher-Order Regularity
    - Korn Inequality
  - Galerkin Methods
    - Well-Posedness
    - Assumptions
    - W_2^m Error Estimates
    - Convergence for Rough Problems
    - W^0 Error Estimates
    - Negative Norm Estimates
    - Non-Coercive Weak Forms
    - Max Norm Error Estimates
- Finite Element Implementations
  - What is Missing
  - Finite Element Assumptions
  - Piecewise Polynomial Approximation
    - Bramble-Hilbert Lemma
    - Lagrange Polynomial Interpolation
    - Non-Smooth Functions
  - Conforming Finite Element Spaces
    - Sufficient Conditions
    - Linear Maps
    - Nonlinear Maps
      - Triangles
      - Quadrilaterals
      - Hexahedra
      - Tetrahedra
      - Wedges
  - Useful Approximations
    - Numerical Quadrature
    - Domain Approximation
    - Essential Boundary Conditions
    - Natural Boundary Conditions
  - Computational Issues
  - Refinement
  - Condition Number Estimates
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