von Neumann Stability Analysis of
Semi-Implcit Flow Equations
The figure below indicates the notation that I will use for spatial location
subscripts. For this example the length of a volume x is assumed to be
constant.
j-3/2
j-1/2
j-2
j+1/2
j
j-1
j+1
The 1-D Euler equations used here are:
 
  V   0
 t x
e 
V
  eV   p
0
 t x
x
V
V 1  p
V

0
t
x x
The Semi-Implicit, Finite Volume method approximating these equations is:
 nj  1   nj
t

 nj  1/ 2V jn11/ 2   nj  1/ 2V jn11/ 2
x
 nj  1e nj  1   nj e nj
t
V jn11/ 2  V jn 1/ 2
t

V
0
 nj  1/ 2e nj  1/ 2V jn11/ 2   nj  1/ 2e nj  1/ 2V jn11/ 2
n
j  1/ 2
x
V jn  V jn 1
x

1
p nj  1  p nj 11
 nj  1/ 2
x
 p nj
V jn11/ 2  V jn11/ 2
x
0
0
Express the edge values of density and specific internal energy as:
 nj  1/ 2   nj  (1   ) nj  1
enj  1/ 2  enj  (1   )enj  1
Express the center value of velocity as:
V jn  V jn 1/ 2  (1   )V jn 1/ 2
Now it’s time for the basic replacements for the von Neumann stability analysis.
Assume that each variable is the sum of a steady solution to the difference
equations, and a perturbation. As before we only look at one representative term
of the Fourier Expansion of the perturbation.
 nj  o   n eik ( j  1)  x
e nj  eo  e n eik ( j  1)  x
p nj  po  p n eik ( j  1)  x
V jn 1/ 2  Vo  V n eik ( j  1)  x
Here to simplify later algebra, I’ve swallowed a factor e-ikx/2 into the definition of V.
Next substitute these relationships into the basic difference equations, cancel out steady
flow equation terms, throw out products with more than one perturbation term, and divide
the whole thing by e-ikx
 nj 1   nj
t

oV n 1 e ikx  1
x
oe nj 1  oe nj
t
V n 1  V n


 nVo (2  1  (1   )e ikx  e  ikx )
x
oe nVo (2  1  (1   )e ikx  e  ikx )
x
Vo V n (2  1  (1   )e ikx  e  ikx )

0
poV n 1 e ikx  1
x
0
p n 1 1  e  ikx 


0
t
x
o  x
We need to introduce an equation of state to eliminate one variable. Write:
p 
p
p
 
e
 e
e 
Grouping terms gives:
 nj  1  
to ik x
 tVo

e  1 V n  1  1 
( 2  1  (1   )eik x  e ik x )  n  0
x
x


e nj  1  
t  po  ik x
 tVo

n 1
( 2  1  (1   )eik x  e ik x ) e n
  e  1 V  1 
x  o 
x





V n 1  1 



 p

tVo
t
p

( 2  1  (1   )eik x  e ik x ) V n 
e ik x  1 
 n  1 
e n  1 
x
xo o
e 

  e

Define two complex coefficients:


t ik x
e 1
x
tVo
C2  1 
( 2  1  (1   )eik x  e ik x )
x
C1 


The matrix form of the equation becomes:



1

0


  C1* p

 o  e
0
1
 C1* p
o e 


C1o    n  1   C
  2
p 
C1 o   e n  1    0
o   n  1  
V   0

1 

0
C2
0
0    n 


0   e n 

C2   V n 
The eigenvalues for the amplification matrix are:
1  C2
2  C2
3  C2

1 



 p
po p 
*
 C1C 
 
 2
1



e



o
e

 p
po p 
 C1C *
1  
 2
1


e



o
e

1 




 C1C * 
1



 p
po p 
 C1C *
1  
 2
1
  e  o e  
 p
po p
 
 2
  e  o e
A review of thermodynamics will tell you that the expression inside the square root is the
derivative of pressure with respect to density at constant entropy, which is the adiabatic
sound speed.
1  C2
Vsound  t


2  2 cos( k x ) 
1  i  x


2  C2
2
2
 V sound  t

2  2 cos( k x ) 
1 

2
x


Vsound  t


2  2 cos( k x ) 
1  i  x


3  C2
2
2
 V sound  t

2  2 cos( k x ) 
1 

2
x


Hence:
1 *  C2 C2*
1
2 *2  3 *3 
C2 C2*
2
 V sound
t2
2
2  2 cos( k x ) 
1 

2
x


Inspecting these results, the first eigenvalue is most restrictive. This is a bit of a mess
for general values of . Looking at the specific case of first order upwind (=1), and
defining Cm to be the material Courant number (Vot/x), we get:
C2 C2*  1  Cm (1  Cm )(2  2 cos( k x))
Which is only less than or equal to 1 when Cm1, the same thing we saw in stability
analysis of a pure advection equation.
For Leith’s method:

1
1  Cm 
2
C2 C2*  1  Cm2 (1  Cm2 )1  cos( k x)
2
Which is also only less than or equal to 1 when Cm1,
For central difference it is unstable:
1
2
*
C2 C2  1  Cm2 1  cos2 ( k x )
