Please help me to understand the following: if I have a transport equation $u_t+au_x=0$ and I want to solve it using finite differences I can see a lot of info on the explicit with central differences and the fact that it is unstable and one sided differences stable provided CFL condition is satisfied. The proofs are via amplification factor. however, if I do Crank Nicolson with central differences, my amplification factor is 1, so I have second order method which is unconditionally stable. Why people don't use or what are the drawbacks of using CN with transport? thanks so much for any help on it!
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The CN method gives extreme oscilations if the transfered impulse isn't smooth. Say, if you prescribe $u(0,t)=C_1$, i.e. a Dirichlet condition, and an inictial condition $u(x,0)=C_2<C_1$ the whole thing would be a big mess. Ideally it shoult be a vertical line travelling across the domain, but in reality you end up using upwind methods, which introduce numerical diffusion and smear the solution.