A BD-entropy inequality preserving Finite Volume scheme for the Quantum Navier–Stokes system

In this work, we propose a Finite-Volume scheme on cartesian grids for the numerical approximation of some solutions of the Quantum Navier–Stokes (QNS) equations. An augmented formulation is implemented by introducing an additional drift velocity v , which reduces the third-order quantum operator to a second-order operator. A set of QNS equations is thus obtained for the density ρ , the drift velocity v = ∇ log ρ and the so-called effective velocity w = u + ν v , where u is the fluid velocity and ν its kinematic viscosity. This formulation includes both the case ν = 0 (Quantum Euler equations) and ν > 0 . The originality of our contribution lies in proposing a scheme that provides some discrete BD-entropy inequalities, similarly to the continuous case, for the system in ( ρ , w , v ) . To do this, we consider a numerical scheme based on a time splitting strategy. The first step consists of a classical first-order Finite-Volume scheme for the hyperbolic part. The second step includes second order terms and source terms, some of which must be carefully discretized. This step is treated implicitly but only requires the inversion of a linear system. Discrete BD-entropy inequalities are established in dimension d . Moreover, for d = 1 , this result is extended to so-called κ -entropies. Numerical tests in one and two dimensions illustrate the efficiency of the scheme in several configurations.