Abstract

It has been known for decades that the probability density functions (PDFs) for solutions of the random/stochastic incompressible Navier-Stokes equations can be represented by a hierarchy of linear equations, but an analogue for compressible flows is still lacking. Moreover, due to the high dimensionality of multipoint PDFs, it is crucial to develop numerical approximations for truncated versions of these hierarchies. As a first step towards this goal, we consider the viscous scalar conservation law with random initial data. This model problem mimics, at the scalar level, the interplay of nonlinear hyperbolic transport and viscosity operators. For this problem, we derive several new types of hierarchies of linear equations for the PDFs of the solution (one of which is the analogue to the incompressible NSE counterpart). The key difference is that, unlike incompressible flows (with the divergence-free condition), the convection term is nonlocal for the PDF master equations in the compressible-like cases. An important observation is that the cumulative distribution function (CDF) of the solution can avoid this nonlocality, and the governing equation contains a term manifesting the unresolved viscous effects that generate the hierarchies. We show that this term can exhibit a ‘dissipative anomaly’ as it does not vanish in the inviscid limit. We further demonstrate that one can approximate this term with sampled solutions to the original scalar conservation law and then use the approximated term to solve the CDFs (and PDFs). The feasibility of this approach is supported both theoretically and numerically.