First, let's say we have a Cauchy problem:
$$ (1) \hspace{0.5cm} u_t (x,t)+ div f(u(x,t))=0, $$
where the initial condition is given with $u(x,0)=u_0(x)$, $x \in A \subseteq \mathbb{R}^d, d \geq 1$, $t \in [0,T]$ and $u \in \mathbb{R}^n, n \geq 1$.
If this system is endowed with the coordinate system of Riemann invariants, we could write it as a diagonal system where the new variables are Riemann invariants
$$ (2) \hspace{0.5cm} s_t (x,t)+ D(s) \cdot s_x(x,t)=0, $$
with initial condition $s_0(x)$ and a diagonal matrix $D(s)$.
And let's take a special kind of initial conditions (called Riemann):
$$(3) \hspace{0.5cm} (u)(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x\geq 0, \end{cases}$$
i.e. in the Riemann invariants form:
$$(4) \hspace{0.5cm} (s)(x,0)= \begin{cases} s_l, x<0 \\[2ex] s_r, x\geq 0, \end{cases}$$
where $u_l$, $u_r$, $s_l$, $s_r$ are constants. From the literature, depending of the concrete system, we know that the solution of $(1),(3)$ consists of shock waves, rarefaction waves and contact discontinuities. This solution is weak solution in the PDE sense.
But what could we say about the solutions of system $(2),(4)$?
Things I know (probably):
- System (2) is not given in the conservative form so we do not have weak solutions in the PDE sense.
- On the other hand, if we know the shocks, rarefactions and contact discontinuities from $(1),(3)$, we could change variables from $u$ to $s$ and get the shocks, rarefactions and contact discontinuities given in the Riemann invariants $s$. But I think that those are not the weak solutions of $(2),(4)$ then.
- If we instead of $(3)-(4)$ have smooth initial conditions, then if we know the strong PDE solutions of $(1)$, we know the strong PDE solutions of $(2)$. But we do not know nothing about the connection between weak solutions of the two problems.
Could we say anything else about solutions of the systems? And are there some other kind of solutions?
I am trying to understand why the system given in Riemann invariants is so important. The only thing I have found so far is that in the system of Riemann invariants rarefaction waves are streight lines parallel to the coordinate axis (page 313 of [Dafermos book]). This is nice, but I hope it is not all.
The Riemann invariant form is mostly important for computing classical solutions such as rarefaction waves and simple waves, along which one Riemann invariant is constant. Rewriting the original problem in the Riemann invariant does not prevent to use the method of characteristics. For instance, this technique can be successfully applied to solve this typical problem. For shocks and contact discontinuities, the solution follows Hugoniot loci. That is to say, a weak solution must satisfy the Rankine-Hugoniot conditions. There is no big advantage of working with Riemann invariants in that case.