The numerical scheme whe have chosen to study the equation:Back I-

CRITERION OF NUMERICAL STABILITY

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.

K=0.03The scheme is numericaly stable

K=0.062The scheme is numericaly stable but we can see that some pertubations begin to take place.

K=0.063We can see that the scheme become numericaly instable fot this value of K.

K=0.065The numerical scheme is instable. Indeed, it diverge.

The numerical scheme become instable from K=0.062

Conclusion

K=0.065The scheme is numericaly stable while it is instable for the same value of K but a different value of dt.

K=0.124The scheme is numericaly stable but we can see that some pertubations begin to take place.

K=0.125We can see that the scheme become numericaly instable fot this value of K.

K=0.128The numerical scheme is instable. Indeed, it diverge.

The numerical scheme become instable from K=0.124

Conclusion

K=0.005The scheme is numericaly stable

K=0.01The scheme is numericaly stable but we can see that some pertubations begin to take place.

K=0.0103The numerical scheme is instable.

K=0.0104The numerical scheme is instable. Indeed, it diverge.

The numerical scheme become instable from K=0.01

ConclusionWe 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.