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Welcome to the Web page of Math 337,
Numerical Differential Equations !
-
- Assignment for Friday, 05/04:
- Read Sec. 15.6.
- Answer QSA 22.
- Attempt Problems 7 and 8.
- HW 15 will be due on Monday following the last day of classes.
- Here is the plan of covering Lecture 15:
- 4/27: Secs. 15.1--15.3;
- 4/29: Sec. 15.4;
- 5/2: Sec. 15.5;
- 5/4: Sec. 15.6;
- 5/6: Sec. 15.7.
- Class
notes
- Lecture
0 : Preliminaries.
- Lecture 1
: Simple
Euler method and its modifications.
- Latex file
of this Lecture. Note that instead of (\ref{...}), one can use
\eqref{...}, provided that you have declared
\usepackage{amsmath} in the preamble.
- Lecture
2 : Runge-Kutta methods.
- Lecture
3 : Multistep,
Predictor-Corrector, and Implicit methods.
- R. Tirani and C. Paracelli, "An algorithm
for starting multistep methods"
- The following are Matlab codes for experimenting with the
order of the starting method to obtain a given 3rd-order multistep
method. Run the main code hw3_starting_method.m .
It calls one of the solvers for the simple Euler,
modified Euler, or
midpoint methods, which are available from the Codes
for examples and selected HW prolems link above, and a multistep
solver hw3_AdamsBashforth3.m
. These solvers, in
their turn, call a function fun4_hw3_starting_method.m,
which implemements a particular function f(x,y) in
y'=f(x,y) for which an exact solution is available. Then one can
investigate the order of the multistep method started by a particular
single-step method by varying the step size, i.e. as described in
Problem 4 of HW 1.
- A. Nordsieck, "On numerical integration of ordinary differential equations," [Math. Comp., v. 16, p. 22 (1962)]
(Understanding one of the key steps of this problem requires the
knowledge of the concept of stability, which is covered in Lecture 4.)
- Lecture
4 : Stability analysis of
finite-difference methods for ODEs.
- Plots
of
stability regions of 3 families of methods (copied from the
notes by Prof. H. Riecke of Northwestern
University; see "Useful links" below for more detail). The plot
for a 5th-order Runge--Kutta method can be found in Fig. 2 of the
review article on ODEs for Encyclopedia of Ecology, posted below.
- Lecture
5 : Higher-order ODEs and
systems of ODEs.
- E. Hairer, C. Lubich, and G. Wanner, "Geometric numerical
integration illustrated by the Stoermer/Verlet method," [Acta
Numerica 2003]
- J. Dummer, "A simple
time-corrected Verlet integration method," (Verlet method with
variable step size), posted on http://www.gamedev.net.
- R. Skeel, "Variable step size destabilizes
the Stoermer/Leapfrog/Verlet method," [BIT, v. 33, p. 172 (1993)].
- Lecture
6 : Boundary-value problems
(BVPs): Introduction.
- Lecture
7 : The shooting method for
solving BVPs.
- Lecture
8 : Finite-difference methods
for BVPs.
- Lecture
9 : Concepts behind
finite-element method.
- Lecture 10 : Eigenvalue
problems. (this is a hand-written version, usually not covered in this course)
- L.F. Shampine, J. Kierzenka, and M.W.
Reichelt, "Solving BVPs for
ODEs in MATLAB with bvp4c," preprint (its
published version appears to be J. Kierzenka and L.F. Shampine,
"A BVP solver based on residual
control and the MATLAB PSE," ACM Transactions on Mathematical
Software 27 (3), 299-316 Sept.
2001). The reason I included this reference on par with Lecture notes
is that it contains clear description on how to solve many different
kinds of BVPs using built-in Matlab commands.
- Lecture
11 : Classification of
partial differential equations (PDEs).
- Lecture
12 : The Heat
equation in one spatial dimension:
Simple explicit method and Stability analysis.
- Lecture
13 : Implicit
methods for
the Heat equation.
- Lecture
14 : Generalizations of
the simple Heat equation.
- Lecture
15 : The Heat
equation in 2 and 3 spatial
dimensions.
- Lecture
16 : Hyperbolic
PDEs: Analytical solutions and characteristics.
- Lecture
17 : Method of
characteristics for solving hyperbolic PDEs
Last
updated: April 2022
- Instructions
how to install Matlab on a student-owned computer at UVM. (Just in case:
This link is supposed to open in a new tab or window. If for any reason
it doesn't (in which case it wouldn't open at all), manually instruct it to open in a new
window or tab.)
- As you are going through the installation,
stay tuned to see and note down the License number (or an Activation
key, if for some reason the installer presents it instead of the
License number). See the next bullet for its usage.
- Instead of installing Matlab on your computer, you may run
your programs on Matlab's server online via this link: https://matlab.mathworks.com/. (I would not recommend this aas your main option, but only as a backup in case that something happens to your computer.)
You will need your Mathworks userID and password, which you created
when signing up with Mathworks during Matlab installation process. This
is where you will need to enter the License number (or an Activation
key).
- MATHEMATICA resources
- Follow this
link to install Mathematica
on your computer.
- On that page, read the (very short)
instructions, click on the link "UVM Software Portal" and follow the
directions there. When prompted, you will
need to login with your UVM NetID. And yes, you will need to create a
Wolfram account if you had not created it before (for example, when
taking Calculus).
- On Wolfram's webpage, you are given the
choice between a Download Manager (4GB) or Direct Download (11+GB). I'd
go with the smaller of these files.
- Be prepared that the installation
process
takes about 30+ minutes (or longer if your connection to Internet is
slow). The entire process consists of two stages, which you will be
guided through by the Installer: You first download the installation
package, and then the Installer installs them on your computer.
- On the same page,
notice the link to "tutorial videos" at the end. It is a very friendly
resource,
which will help you if you do not feel comfortable using Mathematica.
For example, Tutorial 5 in that link talks about defining and plotting
functions.
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