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I am trying to find references in the literature that deal with the Stochastic Riemann problem. Let me explain it a bit. On one hand, in the literature it is not hard to find books or papers that deal with the theoretical and numerical aspects of Deterministic Riemann problem, i.e.

$$(1) \hspace{1cm} \begin{cases} u_t+f(u)_x=0 \\[2ex] u(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x>0 \end{cases} \end{cases} $$

Here $u_l , u_r$ are constants. For this problem we could find a lot things. On the other hand, let's say that we add some randomness in the problem (1). For example, let's say we have (Stochastic Riemann problem):

$$(2) \hspace{1cm} \begin{cases} u_t+f(u)_x=g(u)W(t) \\[2ex] u(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x>0 \end{cases} \end{cases} $$

Here u $\in \mathbb{R}^n$ and W(t) is a white noise. In this case we added some randomness in the source term on the right-hand side of the system. Similarly, we could add randomness in initial conditions, in the flux function, in the coefficients,...

My question is this: Does anyone know any reference where I could find more about theory which concerns Stochastic Riemann problems?

The only thing that maters is that it studies Riemann discontinuous initial conditions theoretically and that it has stochastic in any form mentioned above (I am interested in stochastic sources - problem (2) mainly but any other type of added stochastic in the problem could help me).

So far I've found a few papers/books that mention Stochastic Riemann problems and solving them numerically but I haven't found any paper that deals with theory. All papers given bellow deal with random initial conditions and have no source terms:

  1. Poette, Despres, Lucor - Uncertainty quantification for systems of conservation laws, 2009
  2. Tryoen, Le Maitre, Ndjinga, Ern - Roe solver with entropy corrector for uncertain hyperbolic systems, 2010
  3. Bijl, Lucor, Mishra, Schwab - Uncertainty quantification in computational fluid dynamics, 2013

Also if someone knows some additional papers that deal with this problem numerically write it down please.

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