Efficient high order accurate staggered semi-implicit discontinuous Galerkin methods for natural convection problems

In this article we propose a new family of high order staggered semi-implicit discontinuous Galerkin (DG) methods for the simulation of natural convection problems. Assuming small temperature fluctuations, the Boussinesq approximation is valid and in this case the flow can simply be modeled by the incompressible Navier-Stokes equations coupled with a transport equation for the temperature and a buoyancy source term in the momentum equation. Our numerical scheme is developed starting from the work presented in [1, 2, 3], in which the spatial domain is discretized using a face-based staggered unstructured mesh. The pressure and temperature variables are defined on the primal simplex elements, while the velocity is assigned to the dual grid. For the computation of the advection and diffusion terms, two different algorithms are presented: i) a purely Eulerian upwind-type scheme and ii) an Eulerian-Lagrangian approach. The first methodology leads to a conservative scheme whose major drawback is the time step restriction imposed by the CFL stability condition due to the explicit discretization of the convective terms. On the contrary, computational efficiency can be notably improved relying on an Eulerian-Lagrangian approach in which the Lagrangian trajectories of the flow are tracked back. This method leads to an unconditionally stable scheme if the diffusive terms are discretized implicitly. Once the advection and diffusion contributions have been computed, the pressure Poisson equation is solved and the velocity is updated. As a second model for the computation of buoyancy-driven flows, in this paper we also consider the full compressible Navier-Stokes equations. The staggered semi-implicit DG method first proposed in [4] for all Mach number flows is properly extended to account for the gravity source terms arising in the momentum and energy conservation laws. In order to assess the validity and the robustness of our novel class of staggered semi-implicit DG schemes, several classical benchmark problems are considered, showing in all cases a good agreement with available numerical reference data. Furthermore, a detailed comparison between the incompressible and the compressible solver is presented. Finally, advantages and disadvantages of the Eulerian and the Eulerian-Lagrangian methods for the discretization of the nonlinear convective terms are carefully studied.