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

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