High order semi-implicit schemes for viscous compressible flows in 3D

In this article we present a high order cell-centered numerical scheme in space and time for the solution of the compressible Navier-Stokes equations. To deal with multiscale phenomena induced by the different speeds of acoustic and material waves, we propose a semi-implicit time discretization which allows the CFL-stability condition to be independent of the fast sound speed, hence improving the efficiency of the solver. This is particularly well suited for applications in the low Mach regime with a rather small fluid velocity, where the governing equations tend to the incompressible model. The momentum equation is inserted into the energy equation yielding an elliptic equation on the pressure. The class of implicit-explicit (IMEX) time integrators is then used to ensure asymptotic preserving properties of the numerical method and to improve time accuracy. High order in space is achieved relying on implicit finite difference and explicit CWENO reconstruction operators, that ultimately lead to a fully quadrature-free scheme. To relax the severe parabolic restriction on the maximum admissible time step related to viscous contributions, a novel implicit discretization of the diffusive terms is designed. A variational approach based on the discontinuous Galerkin (DG) spatial discretization is devised in order to obtain a discrete cell-centered Laplace operator. High order corner gradients of the velocity field are employed in 3D to derive the Laplace discretization, and the resulting viscous system is proven to be symmetric and positive definite. As such, it can be conveniently solved at the aid of the conjugate gradient method. Numerical results confirm the accuracy and the robustness of the novel schemes in the challenging stiff limit of the governing equations characterized by low Mach numbers.