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Stability and Amplification FactorI 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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