Analysis of time varying errors in quadratic finite element approximation of hyperbolic problems

Abstract

A methodology for analysing the numerical errors generated by schemes using high-order approximation is presented. Based on Fourier analysis, this methodology is illustrated through the study of the 0-weighting Taylor-Galerkin finite element model applied to an unsteady one-dimension advection problem with quadratic elements. Results show that the dissipation and dispersion errors may be computed by considering simultaneously the so-called physical and computational modes and then, contrarily to what is shown when linear approximation is considered, these errors present a transient behaviour. Moreover, it appears that the errors computed at the end node and at the middle node present in general a different behaviour which in some cases may be opposed to one another. Numerical tests are presented to support the validity of the proposed strategy. We recommend strongly the use of this method for studying the behaviour of numerical schemes based on high-order approximations.

An Analysis of the Discontinuous Galerkin Method for Wave Propagation Problems

“…For hyperbolic problems, Shakib and Hughes [30] provide a Fourier analysis of the space-time Galerkin/least-squares method. The analysis of the Taylor-Galerkin method in one dimension has been proposed by Khelifa and Ouellet [18] and Chaffin and Baker [8]. In these methods, the space discretization is continuous so the inversion of a global mass matrix is generaly required.…”