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hyd January 28, 2010 18:05

Stability and Amplification Factor
 
I have a set of equations where the equations can be written in the following format; dU/dt+A*dU/dx=0 (these are partial differential)
If I discretize this with explicit FTCS
(Ui^n+1)-(Ui^n)=-(A*Delta_t/(2*Delta_x))*((Ui+1^n)-(Ui-1^n)) and try to find out the stability limit, I employ the Von Neumann stability analysis.

For single equations all there is that needs to be done is find G and then find the stability limit from |G|^2<1 (smaller than or equal to), but in this case there is a set of equations which complicates the matter. Let's say
A is a 2x2 matrix
A=[ a b
c a]
Now what needs to be done is finding the eigenvalues and what I end up with is; (I=imaginary)
eigen1=(1-(Delta_t/Delta_x)*((b*c)^0.5)*sintheta)-((Delta_t/Delta_x)*I*a*sintheta)
eigen2=(1+(Delta_t/Delta_x)*((b*c)^0.5)*sintheta)-((Delta_t/Delta_x)*I*a*sintheta)

The stability limit is |eigen*conjugateeigen|<1 (smaller than or equal to 1)
eigen1*conjugateeigen1=(1-(Delta_t/Delta_x)*((b*c)^0.5)*sintheta)^2+((Delta_t/Delta_x)*a*sintheta)^2

The answer is unconditionally unstable for sintheta=-1, but what about sintheta=1. The result should still be unconditionally unstable but it doesn't seem like it.
With sintheta=1
eigen1*conjugateeigen1=(1-(Delta_t/Delta_x)*((b*c)^0.5))^2+((Delta_t/Delta_x)*a)^2 where this can be smaller than 1.
Where am I going wrong?
Thank you.


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