Spectral Analysis of High Order Continuous FEM Hyperbolic PDEs for one and two dimensional problems

In this presentation, we study various continuous finite element discretization
for one and 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 (DeC). The last one allows to alleviate
the computational cost as the mass matrix inversion is replaced by the high
order correction iterations.
To understand the effects of these different combinations, we propose both timecontinuous
and fully discrete Fourier analysis. These allow to compare all of
them in terms of accuracy and stability, as well as to provide suggestions for
optimal values discretization parameters involved. The results are thoroughly
verified numerically both on linear and non-linear problems, and error-CPU
time curves are provided.
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.