  In this work we study various continuous finite element discretization for
two dimensional hyperbolic partial differential equations, varying the
polynomial space (Lagrangian on equispaced, Lagrangian on quadrature points
(Cubature) and Bernstein), the stabilization techniques (streamline-upwind
Petrov-Galerkin, continuous interior penalty, orthogonal subscale
stabilization) and the time discretization (Runge-Kutta (RK), strong stability
preserving RK and deferred correction). This is an extension of the one
dimensional study by Michel S. et al J. Sci. Comput. (2021), whose results do
not hold in multi-dimensional frameworks. The study ranks these schemes based
on efficiency (most of them are mass-matrix free), stability and dispersion
error, providing the best CFL and stabilization coefficients. The challenges in
two-dimensions are related to the Fourier analysis. Here, we perform it on two
types of periodic triangular meshes varying the angle of the advection, and we
combine all the results for a general stability analysis. Furthermore, we
introduce additional high order viscosity to stabilize the discontinuities, in
order to show how to use these methods for tests of practical interest. All the
theoretical results are thoroughly validated numerically both on linear and
non-linear problems, and error-CPU time curves are provided. Our final
conclusions suggest that Cubature elements combined with SSPRK and OSS
stabilization is the most promising combination.
