MA557/MA578/CS557 Numerical Partial Differential Equations

1
MA557/MA578/CS557Numerical Partial Differential
Equations

• Spring 2002
• Prof. Tim Warburton
• timwar_at_math.unm.edu

2
Class and Lab Schedule

• Class
• ESCP 109
• Monday, Wednesday, Friday
• 1000am to 1050am
• Office hours
• By appointment
• -- OR --
• Room 435, Humanities Building
• Tuesday, Thursday
• 200pm 300pm

3
Grade Distribution

• 10 class attendance and participation
• 50 homework assignments
• 40 project work

4
Course Material

• Notes will be available after every lecture..
• Finite Volume Methods for Hyperbolic Problems,
  Randall J. Leveque, Cambridge University Press
• Other materials covered will be supplemented with
  handouts available at
• http//www.useme.org/MA578.html
• I will post this material as promptly aspossible
  after the class.

5
Attendance Policy

• I will endeavor to make this course as
  interactive as possible.
• Most of the ground covered will be accompanied by
  class demonstrations.
• It is strongly recommended that you attend all
  classes.

6
Minimal Homework and Project Presentation
Standards

• All homework handed in must comply with the
  following format
• Student name, top left hand corner of every page
• All sheets of paper must be stapled
• All homework must be typed (I.e. use Word or
  Latex)
• Math symbols may be inserted by hand
• Structure of work must be
• 1) Introduction (description of homework problem
  or project)
• 2) Results including graphs, images and diagrams
• 3) Discussion
• 4) Computer code print outs

Graphs of results are easier to read than large
tables of data
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Coding Comments

• All homework and project codes may be written in
  Matlab, C, C, F77 and even F90/F95
• No support for any other language will be given.
• I strongly suggest you use Matlab for most
  homeworks unless otherwise directed.

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Note

• Note qualified students with disabilities
  needing appropriate academic adjustments should
  contact me as soon as possible to ensure your
  needs are met in a timely manner. Handouts are
  available in alternative accessible formats upon
  request.

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Syllabus

• Week 1 (01/22/03, 01/24/03)
• Introduction to partial differential equations
  and their use.
• Examples of some applications for PDEs
  (acoustics, electromagnetics, fluid dynamics
  .. )
• Review of some basic notation and definitions
  for multivariate calculus.
• Inner-products, norms, Sobolev spaces.

10
Syllabus

• Week 2 (01/27/03, 01/29/03, 01/31/03)
• Derivation of the first order advection
  equation
• Description of characteristic curves in
  time/space.
• Initial conditions, boundary conditions and
  solutions.
• Finite volume solution and computational
  implementation.
• Time stepping methods.
• Testing for accuracy and stability of this
  numerical method.
• Casting the finite volume method as a finite
  difference method.

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Syllabus

• Week 3 (02/03/03, 02/05/03, 02/07/03)
• Introducing the discontinuous Galerkin (DG)
  method.
• Brief review of 1D polynomial interpolation.
• Jacobi polynomials
• Legendre polynomials
• Lagrange interpolating polynomials
• Constructing an arbitrary order DG method for
• Experimental testing of accuracy and stability.
  

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Syllabus

• Week 4 (02/10/03, 02/12/03, 02/14/03)
• Inverse inequalities demonstrating equivalence
  of certain norms (and semi-norms) on hp-type
  finite-element spaces.
• Proof of stability for the DG scheme for
• Proof of consistency for the DG operator.
• Energy based convergence proof for the DG
  method.
• Experimental verification for h, p and T
  dependence of error.
• Explanation of terms in the error estimate.

13
Syllabus

• Week 5 (02/17/03, 02/19/03, 02/21/03)
• Treatment of systems of hyperbolic linear first
  order PDEs.
• Derivation of the 1D acoustic equations.
• DG scheme for the 1D acoustic equations.
• Stability and accuracy for the DG scheme.
• Mini-project each student will implement a
  different system or hyperbolic linear first
  order PDEs using a DG scheme derived from
  scratch.

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Syllabus

• Week 6 (02/24/03, 02/26/03, 02/28/03)
• Derivation of the advection diffusion equation
• Derivation of the local DG (LDG) discretization
  for the second order diffusion term.
• Stability, consistency and convergence.
• Error estimates. Introducing mesh dependent DG
  norms to obtain optimal error estimates.
• Introduction to the Baumann-Oden-Babuska method,
  the Bassi-Rebay method and the interior penalty
  method. General framework will be discussed
  connecting these different approaches.

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Syllabus

• Week 7 (03/03/03, 03/05/03, 03/07/03)
• Stepping up to two-dimensions.
• Building finite-element meshes (introduction to
  existing software and some basic
  constraints/guidelines).
• Determining connectivity of elements in a mesh.
  For DG we need to find the neighboring
  elements of all elements.
• Transforming the physical elements (arbitrary
  triangles) to a reference element. Chain rule
  differentiation and map Jacobian.
• Orthonormal basis for the reference triangle.
  Interpolation on the reference triangle.
  Interpolation error estimate. hp-finite element
  inverse inequalities. Trace inequality.

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Syllabus

• Week 8 (03/10/03, 03/12/03, 03/14/03)
• Construction of mass matrices and
  differentiation matrices for integrating and
  differentiating polynomial functions defined on
  the triangle reference element.
• Matrix conditioning issues for these operators.
• Derivation of TM Maxwells equations
• Boundary conditions.
• DG discretization of the right hand side.
• Consistency/stability/convergence.

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Syllabus

• Week 9 (03/17/03, 03/19/03, 03/21/03)
• NO CLASSES

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Syllabus

• Week 10 (03/24/03, 03/26/03, 03/28/03)
• Examining the eigenvalues of the discrete
  operator.
• DG differential calculus.
• Weak divergence free condition.
• Generalization of DG method to variable material
  properties.
• Perfectly matched layer domain truncation.
  Estimates of effectiveness.
• High-order, asymptotically exact far field
  domain truncation.

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Syllabus

• Week 11 (03/31/03, 04/02/03, 04/04/03)
• LDG/BOB/BR/IP schemes for solving
  with Neumann and Dirichlet boundary conditions.
• Differences between schemes and their
  approximation properties.
• INDIVIDUAL Project building and testing your
  own DG
  Poisson solver.

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Syllabus

• Week 12 (04/07/03, 04/09/03, 04/11/03)
• Completion of INDIVIDUAL Project building
  and testing your own DG Poisson solver.
• Completeness and innovation in testing will be
  strongly rewarded. i.e. push your code as hard
  as possible using
• singular solutions
• testing eigenvalue properties
• discontinuous boundary conditions
• strongly discontinuous material properties
• Generalization to the parabolic heat equation
  with high-order time discretization (ESDIRK
  time stepping).

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Syllabus

• Week 13 (04/14/03, 04/16/03, 04/18/03)
• INDIVIDUAL project presentations.
• Use Powerpoint no exceptions.
• Cover theory, experimental tests, and
  analysis of your results.
• You will have 20 minutes (no more and no less).
• Prepare and rehearse your talk before hand. Make
  sure the talk is structured and
  comprehensible. Do not worry about what your
  fellow students will have covered already.

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Syllabus

• Week 14 (04/21/03, 04/23/03, 04/25/03)
• Introducing the 2D Euler equations for inviscid,
  compressible fluid flow.
• Focus on upwind treatment of boundary conditions
  for all the element interfaces.
• Problems with oscillations in an untreated DG
  solution.
• Filtering.
• Artificial dissipation.
• Using a 2D Euler Matlab DG solver.

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Syllabus

• Week 15 (04/28/03, 04/30/03, 05/02/03)
• Introducing the 2D compressible Navier-Stokes
  equations for inviscid, compressible fluid
  flow.
• Final project (GROUP BASED).
• 1) build an explicit 2D compressible NS solver
  based on the Euler code and the Laplace
  operator already introduced.
• 2) validate and verify
• 3) find limitations by pushing the code to
  breaking point (high mach numbers)

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Syllabus

• Week 16 (05/05/03, 05/07/03, 05/09/03)
• Finish project
• GROUP presentations
• Focus on what went well and what did not. This
  is a demanding application do not worry about
  the theory, just test-test-test.
• I expect conference quality presentations.

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Internet Sources

• http//www.math.umn.edu/cockburn/LectureNotes.htm
  l