NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
Lecture 3
Wednesday, April 6, 2005
Supplementary Reading: Osher and Fedkiw, Section 14.1
Previously we motivated our study of numerical methods for hyperbolic con~
servation laws by focusing on the linear advection equation, t + V r = 0.
We looked at vari
NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
CME 306/MATH 226
solution to homework 3
5.1
Proof of (5.8):
(Ih v)0 (x) v 0 (x) = h1
j
Z
(v 0 (y) v 0 (x) dy = h1
j
Z
Kj
s
Z
h1
j
Z
y
x
Kj
h1
j
Z
y x
1/2
hj v2,Kj
Kj
!
v 00 (t)2 dt dy
Z
(
y
v 00 (t) dt) dy
x
(by CauchySchwarz inequality)
dy
K
NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
CME 306/MATH 226
solution to homework 5
9.3(b)
Boring Version (If you are familiar with matrices, you may skip to the Interesting Version
directly): Let k be the time step, h be the space step in x1 and x2 , and = k/h2 , then
n+1
n
n
n
n
n
n
n
n
Ui,j
Ui,
NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
CME 306/MATH 226
solution to homework 4
5.18
1
The basis is cfw_i,j M
i,j=1 , where i,j attains 1 at (ih, jh) and 0 at other grid points. The support
of each i,j involves six triangles. The function values of at the barycenters are all 1/3. The
gradient a
NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
CME 306/MATH 226
solution to homework 7
Problem 1.
First we extend the initial condition periodically to the whole real line, i.e., we consider the extended
problem
(
ut + (u2 /2)x = 0,
(x, t) R (0, ),
u(x, 0) = 1/2 + sin(x),
x R.
We see that the solution
NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
Lecture 10
Monday, May 2, 2005
Supplementary Reading: Osher and Fedkiw, 18.1
1
Incompressible Flow
Recall the stability condition for compressible ow
max cfw_u + c , u , u
c <
4x
4t
where the quantity on the left of the inequality is the physical wa
NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
Lecture 12
Wednesday, May 11, 2005
Supplementary Reading: CS 205 notes on Poissons equation
1
Laplace Equation
In 1D the Laplace equation is given by
pxx = 0.
The solution is
p = ax + b.
for some constants a and b. In order to nd a and b, we need two boun
NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
Lecture 9
Wednesday, April 27, 2005
Supplementary Reading: Osher and Fedkiw, 14.5.2
1
Discretization
We are interested in constructing a discretization for the hyperbolic system of
conservation laws
~
~ ~
Ut + F U = 0.
(1)
x
~
@F
Assume that the system
NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
Lecture 7
Wednesday, April 20, 2005
Supplementary Reading: Osher and Fedkiw, 14.3.4, 14.4, 14.5
In the last lecture we introduced the ENORoe discretization for evaluating
the numerical ux function. In this lecture we introduce the ENORoe Fix and
ENOLoc
NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
Lecture 11
Monday, May 9, 2005
Supplementary Reading: Osher and Fedkiw, 18.2
In the previous lecture we started looking at incompressible ow, where we
~
have the incompressibility assumption, r V = 0. Under this assumption,
we have a relatively smooth ow,
NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
Lecture 8
Monday, April 25, 2005
Supplementary Reading: Osher and Fedkiw, 14.5.1
In the last lecture we started discussing systems of conservation laws. In particular consider a hyperbolic system of conservation laws with N equations in
one spatial dimens
NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
Lecture 6
Monday, April 18, 2005
Supplementary Reading: Osher and Fedkiw, 14.3.2, 14.3.3
In the previous lecture we introduced the numerical ux function. To review,
we start with the strong form of the conservation law,
ut + f (u)x = 0.
Integrating over a
NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
Lecture 1
Wednesday, March 30, 2005
1
Introduction
This course is concerned with the numerical solution of partial dierential
equations. The focus is on hyperbolic PDE. We will also discuss basic methods for numerically solving parabolic and elliptic equa
NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
Lecture 2
Monday, April 4, 2005
Supplementary Reading: Osher and Fedkiw, Sections 3.3 and 3.5;
Leveque, Sections 6.7, 8.3, 10.2, 10.4
For a reference on Newton polynomial interpolation via divided dierence tables, see Heath, Scientic Computing, Section 7.
NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
Lecture 5
Wednesday, April 13, 2005
Supplementary Reading: Osher and Fedkiw, 14.2; Leveque 4.1, 12.9,
12.10
1
Discrete Conservation Form
To ensure that shocks and other steep gradients are captured by the scheme,
i.e. they move at the right speed even if
NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
CME 306

Winter 2014
CME 306/MATH 226
solution to homework 1
A.2.
L(v) = lim L(v) L(vi ) lim kLkV kv vi kV = 0,
i
i
=
L(v) = 0.
A.5.
1.
Z
Z
2
0
v dx =
dx = 1
1
=
v L2 ().
2.
> 0,
1,
f (x) := 1 x/,
0,
if 1 < x 0,
if 0 < x < ,
if x,
0
f L2 () C (),
kf vk2 .
Thus v can be a