Chapter 1 Hyperbolic Partial Differential Equations
1.1Overview of Hyperbolic Partial Differentia Equations
The One-way Wave Equation
 ut  au x  0

u (0, x)  u 0 ( x)
t  (0, T ), x  
We call this an initial-value problem
We observe that
u(t , x)  u0 ( x  at ) is a solution
Let y  x  at
ut 
ux 

u
y
u
y
 (a)
1
ut  au x   a
u0
u
a 0 0
y
y
We call the lines in (t ,x) s.t.
x  at is constant are characteristics.
a is called the speed of propagation along the
characteristic.
General Hyperbolic Equation
ut  au x  bu  f (t , x)

u (0, x)  u 0 ( x)

where
a , b are constants.
Let
 t
  x  at
Define
u~ ( ,  )  u (t , x) , then
~ u t
u
u x





t 
x 
= ut  a  u x  bu  f ( ,   at )
Therefore
~
u
~  f ( ,   a )
 bu

So

 b
~
u ( , )  u0 ( )e   f ( ,  a )eb(  ) d
0
and
t
u(t , x)  u0 ( x  at )e   f (s, x  a(t  s))eb(t s ) ds
bt
0
Rmk : u (t , x) depends only on values of (t ' , x ' )
s.t. x'at '  x  at i.e , only on the values
of u and f on the characteristic through
(t , x ) for
0  t'  t
System of Hyperbolic Equations
Def:
A system of the form ;
ut  Au x  Bu  F (t , x)
A
u  d
is hyperbolic if the matrix
is diagonalizable with real eigenvalues.
Rmk:
A
is diagonalizable if  a nonsingular
1
matrix P s.t. PAP is a diagonal matrix , that
is ,
 a1
PAP 1  
0

The eigenvalues
0


ad 
of A are the characteristic
ai
speeds of the system.
If B = 0, then
U t  P 1PU x  F
PU t  PU x  PF
( PU )t  ( PU ) x  PF  F
Let
W  PU
Wt  Wx  F
For the case B≠0 ,
Ex:
(see Chapter 9)
ut  2u x  vx  0
vt  u x  2vx  0
u 
 2 1  u 
i.e.    
 
 v   1 2  v 
t
x
As initial data we take
1 i f | x| 1,
u ( x, 0 ) u0 (x) 
0 i f | x| 1,
v( x,0)  0
The system can be written as
(u  v)t  3(u  v) x  0
(u  v)t  (u  v) x  0
using
1 1 
P

1 1
, therefore
w1  u  v  w01 ( x  3t )
w2  u  v  w02 ( x  t )
or
1
1
u (t , x)  ( w1  w2 )  [u0 ( x  3t )  u0 ( x  t )]
2
2
1
1
v(t , x)  ( w1  w2 )  [u0 ( x  3t )  u0 ( x  t )]
2
2
Rmk : The solution consists of two independent
parts, one propagating with speed 3 and one
with speed 1.
Equations with variable coefficients
Consider the equation
ut  a (t , x)u x  0

 u (0, x)  u0 ( x)
Let   t , and  is as yet undetermined , we have
u
x
 ut  u x


Set
Ex:
dx
 a (t , x)  a ( , x)
d
, then
du
0
d
u ( 0 , ) u0 ( )
dx
 a ( ,x )
d
x( 0 ) 
ut  xu x  0
1 if 0  x  1
u ( x,0)  
0 if o.w.
du
0
d
u (0,  )  u0 ( )
dx
x
d
x( 0 ) 
x( )   e
or
  xe t
1.2 Boundary Conditions
Def: The conditions relating the solution of the
differential equation to data at a boundary are
called boundary conditions.
The problem of determining a solution to a
differential equation when both initial data
and boundary data are present is called an
initial-boundary value problem.
Ex:
ut  au x  0
with
0  x  1, t  0
If we specify initial data u(0, x)  u0 ( x) and
boundary data u(t,0)=g(t), then the solution is
given by
 u ( x  at )
u (t , x)   0
1
 g (t  a x)
if x  at  0
if x  at  0
There will be a jump discontinuity in u if
not equal to g(0).
u0 (0) is
1.3 Introduction to finite Difference Schemes
Let h, k  0
then
(t n , X m )  (nk, mh)
and
umn  u(t n , X m )
The set of points (t n , X m ) for a fixed value of n is
called grid level n.
u
u ((n  1)k , mh)  u (nk , mh) (forward
(nk , mh) 
t
k

difference)
u ((n  1)k , mh)  u ((n  1)k , mh) (central difference)
2k

vmn1  vmn
vmn 1  vmn
a
0

k
h

vmn1  vmn
vmn  vmn 1

a
0

k
h

vmn1  vmn
vn  vn

 a m1 m1  0

k
2h
n 1
n 1
n

vm  vm
vm1  vmn 1

a
0
2k
2h

 v n1  1 (v n  v n )
m 1
m 1
vmn 1  vmn 1
 m
2
a
0

k
2h
The forward –time forward-space scheme (1.3.1)
can be written as

n 1
m
 (1  a ) m  a m 1
n
n
where

k
h
Def: We call this a one-step scheme.
The leapfrog scheme (1.3.4) is an example of a
multi step scheme.
Ex:
ut  u x  0
on  2  x  3, 0  t
initial data
1  x if x  1
u0 ( x )  
 0 if x  1
u0 (2)  0
Lax-Friedrichs scheme
  0.8 , h  0.1,

n 1
M
 M 1 , t  1.6
n 1
  1.6 , t  0.8
The leapfrog scheme
  0.8
1.4 Convergence and Consistency
Q: Is vn , m  u (t n , X m ) as k , h  0
Def: Given a partial differential equation
Pu  f , and a finite difference scheme,
Pk , h v  f , we say the finite difference
scheme is consistent with the partial
differential equation if for any smooth
function  (t , x)
P  Pk , h  0 as k , h  0
the convergence being pointwise
convergence at each grid pt.
Ex: The forward-time forward-space Scheme


So
a
t
x
P  t  a x
Let P 
Pk , h 
mn 1  mn
mn 1  mn
a
k
h
where mn   (nk , mh)
by Taylor series we have
1
2
1
 mn  h x  h 2 xx  O(h 3 )
2
mn 1  mn  kt  k 2tt  O(k 3 )
mn 1
So
1
1
Pk , h  t  a x  ktt  ah xx  O(k 2 )  O(h 2 )
2
2
Thus
1
1
P  Pk , h   ktt  ah xx  O(k 2 )  O(h 2 )  0
2
2
as (k , h)  0
 This scheme is consistent
Ex: The Lax-Friedrichs Scheme
1
mn 1  (mn 1  mn 1 )  n   n
2
Pk , h 
 a m 1 m 1
k
2h
by Taylor series
1
2
1
6
mn 1  mn  h x  h 2 xx  h 3 xxx  O(h 4 )
1 n
1
(m 1  mn 1 )  mn  h 2 xx  O(h 4 )
2
2
mn 1  mn 1
1
  x  h 2 xxx  O(h 4 )
2h
6

1
1
1
Pk , h  t  a x  ktt  k 1h 2 xx  ah 2 xxx
2
2
6
 O(h 4  k 1h 4  k 2 )
So Pk , h  P  0 as h, k  0 and
k 1h 2  0
Consistency

The solution of pdes is an
approximation solution of the FDS
Convergence

The solution of FDS approximation
a solution of p d e s
Ex: ut  u x  0 with
vmn 1  vmn vmn 1  vmn

0
k
h
k
 vmn 1  vmn  (vmn 1  vmn )  (1   )vmn  vmn 1
h
k
where  
h
We have shown the scheme is consistency.
Now , let
1 if  1  x  0
u 0 ( x)  
0 elsewhere

1 if  1  mh  0
and v  
0 elsewhere

0
m
but vmn  0 for m  0 , n  0
Therefore
1.5 Stability
If vmn  v so | vmn | has to be bdd is some sense.
Def: A FDS Pk , h vmn  0 for a first-order equation
is stable in a stability region 
If  an integer J s.t.  T>0  CT s.t

J

h  | v |  CT h   | vmj |2
m  
n 2
m
j  0m  
If we let

1
2 2
|| w || h  (h  | wm | )
m  
Then we have
J
1
2 2
h
|| v n || h  (CT  || v j || )
j 0
or
* J
|| v || h  CT  || v j || h
n
j 0
i.e , The norm of the solution at any time
0  t  T , is limit in the amount of growth
that can occur. Note that J might be 0
A stability region is any bounded nonempty
2
region of the first quadrant of  that has
the origin as an accumulation point.
Ex: vmn 1  vmn  vmn 1
We show this is stable if |  |  |  | 1
Pf`:


m  
m  
n 1 2
n
n
2
 | vm |   | vm   vm 1 |

  |  |2 | vmn |2  2 |  ||  || vmn || vmn 1 |  |  |2 | vmn 1 |2
m  

  |  |2 | vmn |2  |  ||  | (| vmn |2  | vmn 1 |2 )  |  |2 | vmn 1 |2
m  

  (|  |2 2 |  ||  |  |  |2 ) | vmn |2
m  

 (|  |  |  |) 2  | vmn |2
m  