Riemann problem for a linear system I want to know how I can determine the type of discontinuity (shock, rarefaction) for a Riemann problem of a linear system such as
$$
q_t+Aq_x=0 .
$$
Let's take the following example:
\begin{align}
A=\left[\begin{array}{ll}
1 & 2 \\
0 & -1
\end{array}\right], \quad q_{l}=\left[\begin{array}{l}
0 \\
1
\end{array}\right], \quad q_{r}=\left[\begin{array}{l}
1 \\
-1
\end{array}\right]
\end{align}
The solution that I found is:
\begin{align}
q(x, t)=\left\{\begin{array}{cc}
q_{l} & x/t<-1 \\
q_{m} & -1<x/t<1 \\
q_{r} & x/t>1
\end{array}\right.
\end{align}
where \begin{align}
q_{m}=\left[\begin{array}{l}
2 \\
-1
\end{array}\right]
\end{align}
Secondly, I can not see what is the difference between the Godunov scheme and a finite volume scheme, are they the seem or Godunov scheme is a finite volume scheme for a Riemann problem.
Finally, what is the initial value for a Godunov scheme, is it the same as the original problem.
Can you please illustrate this on the previous problem?
 A: Set $q=Q(\xi)$ with $\xi=x/t$. We get the eigenvalue problem
$$
A Q' = \xi Q'
$$
showing that either $Q$ is constant, or $\xi \in \{\pm 1\}$ is an eigenvalue of $A$. Therefore, the proposed solution is of the appropriate form. To find the compatible intermediate state $q_m$, we solve the Rankine-Hugoniot system
$$
-(q_m-q_l) = A(q_m-q_l) , \qquad q_r-q_m = A(q_r-q_m)
$$
which is satisfied by the proposed intermediate state. Alternatively, the same result could have been obtained by diagonalization, i.e. by setting $p=R^{-1}q$ where $R$ satisfies $A=R\Lambda R^{-1}$ with $\Lambda = \text{diag}\{\pm 1\}$.
Thus, the proposed Riemann solution with two contact discontinuities is correct -- they are called this way because characteristic speeds are constant accross these discontinuities (and their speed $s=\pm 1$ equals one of the characteristic speeds). They are no shocks since they don't satisfy the Lax entropy condition
$$
\pm 1 > s >\pm 1
$$
where the lhs. is the charactetistic speed on the left of the discontinuity, and the rhs. is the same eigenvalue evaluated on the right of the discontinuity. These waves are no rarefactions either since they are discontinuous.
The Godunov method is a particular finite volume method, and many other numerical methods belong to this family (Lax-Friedrichs, Lax-Wendroff, etc.). Sometimes, finite volume schemes are called Godunov-type methods.
For the derivation of the Godunov method based on Riemann problems, the process involves defining piecewise-constant initial data at time $t_n$, with constant values over a finite volume or elementary cell $\mathcal{C}_i = [x_{i-1/2}, x_{i+1/2}]$. Then, this initial data is evolved in time according to the Riemann solution with discontinuities located at the cell boundaries $x_{i\pm 1/2}$ (for this purpose, a translation to the origin can be performed). Lastly, the solution so obtained is averaged over $\mathcal{C}_i$ at time $t_{n+1}$. See for instance the book by R.J. LeVeque for details (Cambridge University Press, 2002).
