The stochastic fluid-fluid model (SFFM) is a Markov process \(\{(X_t,Y_t,\varphi _t),t\ge 0\}\), where \(\{\varphi _t,{t\ge 0}\}\) is a continuous-time Markov chain, the first fluid, \(\{X_t,t\ge 0\}\), is a classical stochastic fluid process driven by \(\{\varphi _t,t\ge 0\}\), and the second fluid, \(\{Y_t,t\ge 0\}\), is driven by the pair \(\{(X_t,\varphi _t),t\ge 0\}\). Operator-analytic expressions for the stationary distribution of the SFFM, in terms of the infinitesimal generator of the process \(\{(X_t,\varphi _t),t\ge 0\}\), are known. However, these operator-analytic expressions do not lend themselves to direct computation. In this paper the discontinuous Galerkin (DG) method is used to construct approximations to these operators, in the form of finite dimensional matrices, to enable computation. The DG approximations are used to construct approximations to the stationary distribution of the SFFM, and results are verified by simulation. The numerics demonstrate that the DG scheme can have a superior rate of convergence compared to other methods.

随机流体-流体模型 (SFFM) 是一个马尔可夫过程\(\{(X_t,Y_t,\varphi _t),t\ge 0\}\)，其中\(\{\varphi _t,{t\ge 0 }\}\)是连续时间马尔可夫链，第一个流体\(\{X_t,t\ge 0\}\)是由\(\{\varphi _t,t\ ge 0\}\)，第二个流体\(\{Y_t,t\ge 0\}\)由\(\{(X_t,\varphi _t),t\ge 0\} \)。SFFM 平稳分布的算子分析表达式，根据过程的无穷小生成器\(\{(X_t,\varphi _t),t\ge 0\}\), 是已知的。然而，这些算子分析表达式并不适合直接计算。在本文中，不连续 Galerkin (DG) 方法用于以有限维矩阵的形式构造这些算子的近似值，以实现计算。DG近似用于构造SFFM的平稳分布的近似，并通过仿真验证了结果。数值表明，与其他方法相比，DG 方案可以具有更高的收敛速度。