  In this work, we propose a novel strategy for the numerical solution of
linear convection diffusion equation (CDE) over unfitted domains. In the
proposed numerical scheme, strategies from high order Hybridized Discontinuous
Galerkin method and eXtended Finite Element method is combined with the level
set definition of the boundaries. The proposed scheme and hence, is named as
eXtended Hybridizable Discontinuous Galerkin (XHDG) method. In this regard, the
Hybridizable Discontinuous Galerkin (HDG) method is eXtended to the unfitted
domains; i.e, the computational mesh does not need to fit to the domain
boundary; instead, the boundary is defined by a level set function and cuts
through the background mesh arbitrarily. The original unknown structure of HDG
and its hybrid nature ensuring the local conservation of fluxes is kept, while
developing a modified bilinear form for the elements cut by the boundary. At
every cut element, an auxiliary nodal trace variable on the boundary is
introduced, which is eliminated afterwards while imposing the boundary
conditions. Both stationary and time dependent CDEs are studied over a range of
flow regimes from diffusion to convection dominated; using high order $(p \leq
4)$ XHDG through benchmark numerical examples over arbitrary unfitted domains.
Results proved that XHDG inherits optimal $(p + 1)$ and super $(p + 2)$
convergence properties of HDG while removing the fitting mesh restriction.
