A conservative, weakly nonlinear semi-implicit finite volume scheme for the compressible Navier−Stokes equations with general equation of state

摘要

In the present paper a new efficient semi-implicit finite volume method is proposed for the solution of the compressible Euler and Navier−Stokes equations of gas dynamics with general equation of state (EOS). The discrete flow equations lead to a mildly nonlinear system for the pressure, containing a diagonal nonlinearity due to the EOS. The remaining linear part of the system is symmetric and at least positive semi-definite. Mildly nonlinear systems with this particular structure can be very efficiently solved with a nested Newton-type technique.The new numerical method has to obey only a mild CFL condition, which is based on the fluid velocity and not on the sound speed. This makes the scheme particularly interesting for low Mach number flows, because large time steps are permitted. Moreover, being locally and globally conservative, the new method behaves also very well in the presence of shock waves. The proposed algorithm is first validated against the exact solution of a large set of one-dimensional Riemann problems for inviscid flows with three different EOS: the ideal gas law, the van der Waals EOS and the Redlich−Kwong EOS. In the final part of the paper, the method is extended to the two-dimensional viscous case.