  In this paper we present a new high order semi-implicit DG scheme on
two-dimensional staggered triangular meshes applied to different nonlinear
systems of hyperbolic conservation laws such as advection-diffusion models,
incompressible Navier-Stokes equations and natural convection problems. While
the temperature and pressure field are defined on a triangular main grid, the
velocity field is defined on a quadrilateral edge-based staggered mesh. A
semi-implicit time discretization is proposed, which separates slow and fast
time scales by treating them explicitly and implicitly, respectively. The
nonlinear convection terms are evolved explicitly using a semi-Lagrangian
approach, whereas we consider an implicit discretization for the diffusion
terms and the pressure contribution. High-order of accuracy in time is achieved
using a new flexible and general framework of IMplicit-EXplicit (IMEX)
Runge-Kutta schemes specifically designed to operate with semi-Lagrangian
methods. To improve the efficiency in the computation of the DG divergence
operator and the mass matrix, we propose to approximate the numerical solution
with a less regular polynomial space on the edge-based mesh, which is defined
on two sub-triangles that split the staggered quadrilateral elements. Due to
the implicit treatment of the fast scale terms, the resulting numerical scheme
is unconditionally stable for the considered governing equations. Contrarily to
a genuinely space-time discontinuous-Galerkin scheme, the IMEX discretization
permits to preserve the symmetry and the positive semi-definiteness of the
arising linear system for the pressure that can be solved at the aid of an
efficient matrix-free implementation of the conjugate gradient method. We
present several convergence results, including nonlinear transport and density
currents, up to third order of accuracy in both space and time.
