  Introduced by Bean and O'Reilly (2014), a stochastic fluid-fluid process is a
Markov processes $\{X_t, Y_t, \varphi_t\}_{t \geq 0}$, where the first fluid
$X_t$ is driven by the Markov chain $\varphi_t$, and the second fluid $Y_t$ is
driven by $\varphi_t$ as well as by $X_t$. That paper derived a closed-form
expression for the joint stationary distribution, given in terms of operators
acting on measures, which does not lend itself easily to numerical
computations.
  Here, we construct a discontinuous Galerkin method for approximating this
stationary distribution, and illustrate the methodology using an on-off
bandwidth sharing system, which is a special case of a stochastic fluid-fluid
process.
