Finite-difference vs finite-volume schemes for conservation laws

As far as I know we don't use finite difference scheme for conservation law because solution of conservation law makes no sense pointwise as its only in $$L^{\infty}$$. But however we use finite difference scheme for linear transport equation ($$u_t+au_x=0$$), which is a conservation law with flux $$f(u)=au$$. Why is this so? What is the difference between solutions of transport equation and conservation laws when flux is not linear? What happens if we use finite volume schemes for transport equation?

• Finite difference schemes are certainly used for conservation laws and finite volume schemes can absolutely be used for the transport equation. I suspect that the distinction you want to make is rather between problems that admit discontinuous solutions even with smooth initial data and those that do not, but correct me if I'm wrong – ekkilop Feb 9 at 16:27