Transcript MA375 - Rice U - Computational and Applied Mathematics

MA557/MA578/CS557
Numerical Partial
Differential Equations
Spring 2002
Prof. Tim Warburton
[email protected]
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Class and Lab Schedule
• Class:
– ESCP 109
• Monday, Wednesday, Friday
• 10:00am to 10:50am
• Office hours:
– By appointment
-- OR -– Room 435, Humanities Building
• Tuesday, Thursday
• 2:00pm – 3:00pm
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Grade Distribution
• 10 % class attendance and participation
• 50 % homework assignments
• 40 % project work
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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 as
possible after the class.
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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.
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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….
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Syllabus
Week 2 (01/27/03, 01/29/03, 01/31/03)
u
u
a
0
• Derivation of the first order advection equation:
t
x
• 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.
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• Casting the finite volume method as a finite difference method.
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
u
u
a
0
• Constructing an arbitrary order DG method for
t
x
• 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.
u
u
• Proof of stability for the DG scheme for
a
0
t
x
• 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.
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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)
u
u
 2u
a
b 2
• Derivation of the advection diffusion equation:
t
x
 x
• 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 14
approaches.
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
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inverse inequalities. Trace inequality.
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.
H x
Ez

• Derivation of TM Maxwell’s equations:
t
y
H y
E z
• Boundary conditions.

t
x
• DG discretization of the right hand side.
H y H x
E z


t
x
y
• 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:
  u    u 
 a    b   f  x, y 
x  x  y  y 
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
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time discretization (ESDIRK time stepping).
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.html
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