MTH
453-
553
: NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS - Spring 2013
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Assignments |
RECALL that textbook
exercises
are
here.
FORM of homework solutions:
show enough work required to solve a given
problem. If you are asked to write a program, attach a listing of the
code. The results of a program, graphs, and comments need to be
correctly labeled and signed. Please do not overdo: I can't read
100-page essays. The core of the presented work should be your insight
into the nature of the problem. Present your conclusions, not merely a
bunch of graphs, numbers, and/or lines of code collated together.
Important: policy on group work vs individual work. It
is OK for students to talk about problems that they are
solving. However, no exchange of written materials is allowed unless
explicitely stated. In particular, every part of a computing
assignment has to be typed individually by a student. Violation of
these rules will be treated seriously.
Schedule and assignments
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(4/1) Introduction, basic ideas for solving differential equations
with finite differences. IVP and BVPs. Boundary conditions for a
two-point BVP. [Review Chapter 1].
- (4/3)
Continue FD formulation for a two-point BVP. [Read Chapter 2 for background].
Truncation error for two-point BVP. Start Poisson's equation [Chapter 3].
Assignment 1 due Monday, 4/15, in class.
- (4/5) LAB: meet in MLC computer lab Kidd 108.
You can solve and turn in any of the Exercises 1.2, 2.4, 2.5 for extra credit.
Lab is reserved for our class 9:00-11:00. If you need extra time for practice, come earlier.
Log in using your ONID ID to the computer. Start working through lab.txt.
- (4/8). FD formulation for Poisson's problem. How to solve it ?
(linear solvers). Truncation error. [Read Chapter 3 and supplement with Chapter 2]
- (4/10). Error equation and error analysis using vector norms. Stability.
- (4/12). (Minus) discrete Laplacian is positive definite. Eigenvalues
of discrete Laplacian.
- (4/15) General elliptic equations.
- (4/17) An alternative implementation of Poisson's equation:
fd2d.m Boundary conditions other than Dirichlet. [details in Chapter 2.12, 2.15]
- (4/22) Recap: how to implement Neumann conditions for Poisson's equation.
Start heat equation.
Assignment 2 due Friday 4/26. due to power outages last Sunday, HW due Monday, 4/29
. Take advantage of extra credit opportunities !
- (4/24) [Chapter 9] Basic schemes for the heat equation, and expected behavior
of the algorithm and of the error. Local truncation error analysis.
- (4/26) Stability of FE and BE combined with the usual 3-pt stencil.
- (4/28) Von-Neumann stability analysis: FE.
Assignment 3. Theoretical part Pbms 1-2
due Monday 5/6. Computational part Pbms 3-4 due Monday 5/13.
- (5/1) Examples of von-Neumann stability analysis.
- (5/3) Truncation error analysis for an explicit-implicit scheme
for the heat equation. [Worksheet].
- (5/6) Wrap-up Fourier series/transform and connection to
von-Neumann analysis.
- (5/8) Review for Exam. [EXTRA OFFICE HOURS].
- (5/10) Midterm, in class.
- (5/13) Wrap-up elliptic and parabolic PDEs: dissipative character of
the models and of the numerical approximations.
- (5/15) Convergence worksheet (Lax Equivalence Theorem). Lax-Richtmyer
stability.
- (5/17) Start hyperbolic problems. Overview of first order hyperbolic PDEs
and extensions, second order hyperbolic PDEs.
Method of characteristics. [Read Chapter 10].
- (5/20) Schemes for advection equation u_t+au_x=0: explicit and implicit.
FTFS, FTCS, Lax-Friedrichs, Lax-Wendroff. Accuracy order.
- (5/22) Stability of upwind schemes using von-Neumann Ansatz. Plot stability
regions in the complex plane.
- (5/24) NO CLASS TODAY.
(5/27) No CLASS: Memorial Day.
Assignment 4 (theory) due Friday, 5/31.
The computational part (TBA) will be due Friday, June 7.
- (5/29) Method of Lines (MOL) stability analysis of the upwind
scheme. Connection of the amplification factor to the eigenvalues of
the matrix in MOL formulation. Facts for other schemes.
- (5/31) Numerical simulations with advection1d.m. How to
choose k, and factor lambda. How to plot the error and find the order
of convergence.
Diffusive versus dispersive character of schemes (read 10.9) via
modified equation analysis (TBA)>
- (6/3) Modified equation for the schemes for one-way wave equation
u_t+au_x-0.
Schemes for other equations: advection-diffusion,
advection-reaction. System of hyperbolic transport equations versus
two-way wave equation.
- (6/5) No class today. Work on your last assignment due Friday 6/7
in class.
Final exam will be posted here.
- (6/7) Review and wrap-up.
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