On Kinetic Flux Vector Splitting Schemes for Quantum Euler Equations∗

The kinetic flux vector splitting (KFVS) scheme, when used for quantum Euler equations, as was done by Yang et al [22], requires the integration of the quantum Maxwellians (Bose-Einstein and Fermi-Dirac distributions), giving a numerical flux much more complicated than the classical counterpart. As a result, a nonlinear 2 by 2 system that connects the macroscopic quantities temperature and fugacity with density and internal energy needs to be inverted by iterative methods at every spatial point and every time step. In this paper, we propose to use a simple classical KFVS scheme for the quantum hydrodynamics based on the key observation that the quantum and classical Euler equations share the same form if the (quantum) internal energy rather than temperature is used in the flux. This motivates us to use a classical Maxwellian – that depends on the internal energy rather than temperature – instead of the quantum one in the construction of KFVS, yielding a KFVS which is purely classical. This greatly simplifies the numerical algorithm and reduces the computational cost. The proposed schemes are tested on the 1-D shock tube problem for the Bose and Fermi gases in both classical and nearly degenerate regimes.