1.4 Convergence and Consistency
Q: Is vn , m  u (t n , X m ) as k , h  0
Def: A one-step finite difference scheme
approximating a partial differential equation is a
convergent scheme if
For any solution to the partial differential equation
u (t , x) , and solutions to the finite difference
n
0
v
v
scheme, m , such that m converges to u 0 ( x) as
n
v
mh converges to x, then m converges to u (t , x)
as (nk,mh) converges to (t,x) as h,k converge to 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 point.
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
Def: If F and G are function of some parameter
 , we write
F=O(G) as   0,
if
F
K,
G
for some constant K and all
F=o(G) as
 sufficiently small.
  0,
if F/G converges to zero as
 tends to zero.
Examples:
F= O(h r ) means F goes to zero as fast/slow as
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
Ex: ut  u x  0 with
Convergence

The solution of FDS approximation
a solution of p d e s
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 bounded 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
* J
|| v || h  CT  || v j || h
n
or
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.
Example: vmn 1  vmn  vmn 1
We show this is stable if |  |  |  | 1
Proof`:

| v
m  

|   | vmn   vmn 1 |2
n 1 2
m
m  

  |  |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

 | vm |
n
2
m  
How about inhomogeneous problem?
In Chapter 9:
A scheme is stable for the Pk ,h v  f if it is stable
For the equation Pk ,h v  0 .
The Lax-Richtmyer Equivalence Theorem
The importance of the concepts of consistency
and stability is seen in the Lax-Richtmyer
equivalence theorem, which is the fundamental
theorem in the theory of finite difference schemes
for initial value problem.
Theorem: A consistent finite difference scheme
for a pde for which the initial value problem is
well-posed is convergent if and only if it is
stable.
Def: The initial value problem for the first-order
partial differential equation Pu =0 is
well-posed if for any time T>=0, there is a
constant CT such that any solution
u (t , x) satisfies

2



u (t , x) dx  CT  u (0, x) dx
2

for 0  t  T .
Consistency  Convergence
stability
easy, concepts
hard analysis
Remark: Non-stable scheme can not be
convergent.
1.6 The Courant-Friedrichs-Lewy Condition
The CFL Condition
Example 1.5.1
vmn 1  vmn 1  vmn 1 is stable if |  |  |  | 1
i.e. the Lax-Friedrichs scheme is stable
if a  1, where  
k
h
1
vmn 1  (vmn 1  vmn 1 )
n
n
v

v
2
 a m 1 m 1  0
k
2h
1 1
1 1
vmn 1  (  a )vmn 1  (  a )vmn 1
2 2
2 2
|
1 1
1 1
1
1
 a |  |  a | |1  a |  |1  a | 1
2 2
2 2
2
2
Claim: | a | 1 is necessary condition for stability
Theorem: For an explicit scheme for the hyperbolic
equation (1.1.1) of the form
vmn 1  vmn 1  vmn  vmn 1 with
k

h
a necessary condition for stability is the
Courant-Friedrichs-Lewy(CFL) condition
| a | 1
pf: Suppose | a | 1,
Consider u (1,0) , it depends on either u0 (a) or
u 0 ( a )
but v0n depends on x only for | x | 1 | a |
since v0n depends on vm0 for m  n
and mh  nh  n1k  1 | a |
so v0n
u (1,0)
so it is not stable
Theorem: There are no explicit unconditionally
stable, consistent finite difference schemes for
hyperbolic systems of pdes.
Two implicit schemes which are unconditionally
stable
backward-time
center-space scheme
vmn 1  vmn
vmn 11  vmn 11
 a
0
k
2h
This will be showed in Section 2.2
backward-time
backward-space scheme
vmn 1  vmn
vmn 1  vmn 11
 a
0
k
h

(1  a )vmn 1  vmn  avmn 11
(1  a ) 2 | vmn 1 |2 | vmn |2 2a | vmn || vmn 11 | (a ) 2 | vmn 11 |2
 (1  a ) | vmn |2 (a  (a ) 2 ) | vmn 11 |2


(1  a )  | vmn 1 |2
2
m  


 (1  a )  | v |  (a  (a ) )  | vmn 1 |2
m  


| v
m  
n 1 2
m
if a>0
n 2
m

|   | vmn |2
m  
2
m  