§6.2 Parabolic systems and Boundary
Conditions
A system of the form
(*) ut  Buxx  Aux  cu  F (t , x) on 
is parabolic, or Petrovskii parabolic if the
eigenvalues of B all have positive real parts.
A common special case is when B is positive
definite, but in general the eigenvalues need not be
real, nor does B have to be diagonalizable. Notice
that no restriction are placed on matrices A and C.
Theorem: The initial value problem for the system
(*) is well-posed in the sense of Def 1.5.2, and
actually a stronger estimate holds, for each T>0
there is a constant C T
s.t

t 

0 
2
 | u (t , x) | dx    | u x ( s, x) | dxds

t 
 CT (  | u (0, x) | dx    | F ( s, x) |2 dxds)

2
0 
0  t  T
Note: This also gives a bound on the derivative of u
w.r.t x in addition to a bound for u. The bound on
u x implies that the solution is infinitely
differentiable for positive t.
Proof:
Assume F(t,x)=0 (for homogeneous equation)
Let’s consider the Fouries transform of (*)
uˆt  ( w2 B  iwA  c)uˆ
For large values of |w| the eigenvalues of the matrix
 w 2 B  iwA  c
must have real parts that are less than  bw 2 for
some b>0.
Now consider
 w2 B  iwA  c  w2 [ B  i(w) 1 A  (w) 2 c]
 Eigenvalues of a matrix are continuons function
of the matrix.
 the eigenvalues of B  i(w) 1 A  (w) 2 c must
be close to those of B itself.
 the eigenvalues of  w 2 B  iwA  c
have real parts bounded by a  bw 2 for some
b0
, and some value a .
 uˆ (t , w)  e (  w
2 B  iwt  c ) t
uˆ0 ( w)
(by Appendix A,10)
| uˆ (t , w) | Ke ( a  bw

2 )t
| uˆ 0 ( w) |

  | uˆ (t , w) | dw  K t  | uˆ 0 ( w) |2 dw
2


t 
and

2
  w | uˆ ( s, w) | dwds  k t  | uˆ 0 ( w) | dw
2
2
0 

then by Parseval’s relation.
Also the non-homogeneous case follows easily.
Boundary Condition
Dirichlet boundary condition
u  b0
on 
or
Neumann boundary condition
u
 b1
x
on 
or (Mixed)
Robin boundary condition
u  
u
 b2
x
on 
6.3 Finite Difference Scheme for Parabolic Eqns
Forward-time central-space scheme
vmn 1  vmn
vmn 1  2vmn  vmn 1
(*)
b
k
h2
or
vmn 1  (1  2b )vmn  b (vmn 1  vmn 1 )
k
h2
The parameter  plays a role for parabolic
where  
equations similar to the role of  for hyperbolic
equations.
(*) is (1.2) accurate
Let vmn  g n e im
We have
g 1
e i  2  e  i
b
k
h2
or
k i
 i
(
e

e
 2)
2
h
1
 g  1  4b sin 2 
2
g  1 b
| g | 1   b 
1
2
and is disspative of order 2 as long as b 
1
2
Remark: Dissipatively is a desirable property for
scheme for parabolic equation to have since then
the FD solution will become smoother in time as
does the solution of the solution of the pde.
Notice that even though the scheme is accurate of
order (1,2), because of the stability condition the
scheme is second-order accurate if
 is constant.
The backward-time central-space scheme
vmn 1  vmn
vmn 11  2vmn 1  vmn 11
n
b

f
m
k
h2
Implicit, unconditional stable.
Of accurate of order (1.2) and is dissipative when
 is bounded away from 0
The Crank-Nocolson scheme
vmn 1  vmn 1 vmn 11  2vmn 1  vmn 11
 b
k
2
h2
1 vmn 1  2vmn  vmn 1 1 n 1
n
 b

(
f

f
m
m )
2
h2
2
Implicit unconditioned stable.
Accurate of order (2,2)
Dissipative of order 2 if  is constant.
But not dissipative if  is constant.
Notice the actual accuracy may depend on the
smoothness of initial conditions.
The Leapfrog scheme
vmn 1  vmn 1
vmn 1  2vmn  vmn 1
n
b

f
m
2k
h2
and this scheme is unstable for all values of  .