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.