A high-order discontinuous Galerkin method for natural convection problems

Discontinuous Galerkin (DG) methods have proved to be very well suited for the construction of robust high-order numerical schemes on unstructured and possibly non conforming grids for a wide variety of problems. In this paper we consider natural convection flow problems and present a high-order DG method for their numerical solution. The governing equations are the incompressible Navier-Stokes (INS) with the Boussinesq approximation to represent buoyancy effects and the energy equation to describe the temperature field. The method here presented is an extension to natural convection flows of a novel high-order DG method for the numerical solution of the INS quations, recently proposed in [1]. The distinguishing feature of this method is the formulation of the inviscid interface flux which is based on the solution of local Riemann problems associated with the artificial compressibility perturbation of the incompressible Euler equations. The discretization of the viscous term follows the well established DG scheme named BR2 [2, 3]. The method is fully implicit and the solution is advanced in time using either a first order backward Euler or a second order Runge-Kutta scheme. To assess the capabilities of the DG method developed in this paper we computed second-, third-and fourth-order space-accurate solutions of several benchmark problems on natural convection in two-dimensional cavities