An efficient second order all Mach finite volume solver for the compressible Navier–Stokes equations

In the numerical simulation of fluid dynamic problems there are situations in which acoustic waves are very fast compared to the average velocity of the fluid and conversely situations in which the fluid moves at high speed and shock waves may be present. Ideally, a numerical method should be able to treat these different regimes without strong limitations in terms of time step and without excessive related computational cost. Unfortunately, standard explicit in time schemes often adopted for hyperbolic problems are not suitable for these problems, hence remedies have to be studied. To this aim, the results presented in this article concern the development of a second order in time and space numerical method for the compressible Navier–Stokes equation which works for both high and low Mach numbers. In particular, when the Mach number goes to zero, one recovers a numerical method for the limit Navier–Stokes system which under some additional hypothesis degenerates to the incompressible Navier–Stokes equations, while in the case of high Mach numbers the method exhibits a shock capturing structure. The idea is based on partitioning the equations into a fast and a slow scale and by taking implicit the fast scale dynamic together with the viscous terms. The resulting numerical scheme is stable for time steps which are independent both from the speed of the pressure waves and from the diffusive terms characterizing the viscous forces and the heat flux. The only time step limitation is induced by the average speed of the flow. The work here presented extends the seminal ideas developed in Dimarco et al. (2017, 2018) for isentropic Euler equations and in Boscheri et al. (2020) for the full set of compressible Euler equations to the multidimensional Navier–Stokes system and permits efficient three dimensional simulations of all Mach problems. The discretization is constructed on Cartesian meshes and the method is second order accurate in space and time. Numerical results show the accuracy, the robustness and the effectiveness of the new proposed approach.