I'm looking for references to understand where the numerical instabilities come from in hydrodynamics in general, and notably when the Péclet number: $Pe>1$. I'm using the finite element method.

Would you have some references for that, please? Or if you could explain at least the case of the Péclet instability for example in the equation $c_t+u.\nabla c=D\nabla^2c$ in 1d, even in the finite difference method I'd be happy.

I looked at this course and it looks like it comes from numerical observation... Is there any more formal derivation of the instability?

Thank you in advance

Edit: I found this course about the finite volume method. But if there is something that explains it in the context of the finite element method or finite difference method it would be better for me.



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