I am solving a non-linear advection equation of the form $u_t + f(u)_x = 0$ where $f(u)$ is a complicated function of $u$. I am solving this equation using a first order fully implicit scheme (backward Euler in time and upwind for space) with second order linearization using Newton's method for non-linear convergence. In order to analyze the results (whether they make sense physically) I wanted to get an idea of the general characteristics that they may exhibit (like shocks and rarefactions) as a function of the nature of $f(u)$. However, in the literature, every treatment of non-linear advection that I find ultimately ends up dealing with inviscid Burgers equation. While I understand the rationale, I would like to look at some other examples. Specifically, my flux function can be convex or concave i.e., $f^{\prime\prime}(u)$ can be $< 0$ or $> 0$ and I wanted to look at cases where the non-linear wave-speeds $f^\prime(u)$ are something other than $u$ itself. Any relevant reference is much appreciated.


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