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?
2026-03-25 06:10:34.1774419034
Finite-difference vs finite-volume schemes for conservation laws
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For numerical methods up to order two, there is no clear distinction between finite difference methods and finite volume methods if the flux is linear. The main reason for that is that the error between cell averages and point values is itself of second order. For instance, the Lax-Wendroff method is typically a finite difference method, but it can also be interpreted as a finite volume method.