Consistency of a Riemann solver appear in the theory of conservation laws and systems of conservation laws. It is defined as follow:
A Riemann solver $\mathcal{R}$ is consistent if it satisfies the following two properties:
Fix $u_l ,u_r$
For any $u_m$, such that $\exists x \in \mathbb{R}, \mathcal{R}[u_l,u_r](x)=u_m$ we must have $$\mathcal{R}[u_l,u_m](y)=\begin{cases} \begin{align*} \mathcal{R}[u_l,u_r](y) & \, \forall y<x \\ u_m & \, \forall y \geq u_m \end{align*}\end{cases} $$ and $$\mathcal{R}[u_m,u_r](y)= \begin{cases} \begin{align*} u_m & \, \forall y\leq x \\ \mathcal{R}[u_l,u_r](y) & \, \forall y>x \end{align*}\end{cases}$$
If there exists $x$ such that $\mathcal{R}[u_l,u_m](x)=\mathcal{R}[u_m,u_r](x)$ then $$\mathcal{R}[u_l,u_r](y)=\begin{cases} \begin{align*} \mathcal{R}[u_l,u_m](y) & \,\forall y\leq x \\ \mathcal{R}[u_m,u_r](y) & \,\forall y \geq x \end{align*} \end{cases}$$
My question where does this notion play a role, how is it usefull ?