Approximation of a simple Navier-Stokes model by monotonic rearrangement (Q379807)

The one space dimensional Navier-Stokes model for compressible fluid is studied in the paper. NEWLINE\[NEWLINE\begin{aligned} \frac{\partial }{\partial t}(\rho v)+ \frac{\partial }{\partial x}(\rho v^2+p(\rho))- \frac{\partial }{\partial x}\left(\mu(\rho)\frac{\partial v}{\partial x}\right)=0, \quad x\in \mathbb{R},\;t>0 \\ \frac{\partial \rho}{\partial t} +\frac{\partial }{\partial x}(\rho v)=0 ,\quad x\in \mathbb{R},\;t>0,\end{aligned} NEWLINE\]NEWLINE where \(\rho(x,t)>0\) is the density of the fluid, \(v(x,t)\) is the velocity, \(p\) is the pressure, \(\mu\) is the viscosity. It is proposed that NEWLINE\[NEWLINE \mu(\rho)=\varepsilon \rho,\quad p(\rho)=\lambda\varepsilon\rho. NEWLINE\]NEWLINE The parameter \(\varepsilon\) is interpreted as a level of noise.NEWLINENEWLINEThe problem is writing in Lagrangian coordinates. After that the discrete scheme based on the simulation of heat equation by random walk is constructed. It is proved that the interpolation of discrete solution converges to the solution of the Navier-Stokes system. The using method can be applied to the more general problems.