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I-CRITERION OF NUMERICAL STABILITY

The numerical scheme whe have chosen to study the equation:

is an explicit scheme and so, we can see that this scheme is instable                                           for some values of K, dx and dt.

In this section, we fixe the parameters = -2/27, =1,=1 and limits conditions      u(0)=ue3= -1/3+(1/3)^1/2,  u(L)= ue1= -1/3-(1/3)^1/2,

Furtermore, we fixe t=2.0 and L=1.0.

Then, we study the stability of our numerical scheme by fixing dt and dx and schearching the value of K for which the scheme diverge.

1-dt=2.10-² and dx=5.10-²:

The scheme is numericaly stable

The scheme is numericaly stable but we can see that some pertubations begin to take place.

We can see that the scheme become numericaly instable fot this value of K.

The numerical scheme is instable. Indeed, it diverge.

The numerical scheme become instable from K=0.062

2-dt=10-² and dx=5.10-²:

The scheme is numericaly stable while it is instable for the same value of K but a different value of dt.

The scheme is numericaly stable but we can see that some pertubations begin to take place.

We can see that the scheme become numericaly instable fot this value of K.

The numerical scheme is instable. Indeed, it diverge.

The numerical scheme become instable from K=0.124

3-dt=2.10-² and dx=2.10-²:

The scheme is numericaly stable

The scheme is numericaly stable but we can see that some pertubations begin to take place.

The numerical scheme is instable.

The numerical scheme is instable. Indeed, it diverge.

The numerical scheme become instable from K=0.01

4-Conclusion:

We can define a parameter D which control the numerical stabilty of our scheme.

The previous study show us that :

For dt=2.10-² and dx=5.10-², the numerical scheme is stable only if D10.496

For dt=10-² and dx=5.10-², the numerical scheme is stable only if D20.496

For dt=2.10-² and dx=2.10-², the numerical scheme is stable only if D30.5

So, we have shown that our numerical scheme is stable only if D0.5

We find again the result given by the theoretical analysis of numerical schemes.
 
 

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