I'm just curious
Suppose I'm solving
$u_t + au_x=f$
on the interval $x \in [a,b]$. Suppose there is an interface point $\alpha$ that that the solution on the left of this point is $u^+$ and the solution to the left is $u^-$. Additionally based on what side we are on we have different advection speeds i.e on the left $a^-$ and on the right $a^+$. I'm curious if we define $[u] = u^+ - u^- = ?$. What is this? Or can we calculate $[au] =a^+u^+ - a^-u^- =?$. I have calcuated the first and second order jump conditions, just used the pde, I'm have problem deriving the zero order jump conditions. Can we use some sort of weak formulation here?