Question about explicite and implicite time integration theory
 

Hi,

Usually we prefer explicit treatment to the convective term but implicit treatment to the diffusive term. From some papers, I know the reason of this difference is the stability. But I'm wondering how to derive this. How to derive that for stability reason we use different treatment?

The question is more complicate ... the non-linear term, when discretize in time in an implicit way leads to a non-linear algebric system. Then you have to face by using some techniques... (for example you can see the book of Ferziger and Peric).

Generally the diffusive part is linear and is better suited for implicit discretization. Furthermore, the stability constraints are more critical for the the diffusive term (e.g. first order implicit 1D equation has the limit alpha*dt/h^2 < 0.5) than for the linear one. But the Fourier stability analysis give some response for the linear case.

I suggest you to derive the stability regions for the 1D equation du/dt + a du/dx = alpha d2u/dx^2 using implicit and explicit discretizations