A theoretical Taylor-Galerkin model for first-order hyperbolic equation

This paper presents a class of Taylor-Galerkin (TG) finite-element models for solving the first-order hyperbolic equation which admits discontinuities. Five parameters are introduced for purposes of controlling stability, monotonicity and accuracy. In this paper, the total variation diminishing concept and the theory of M-matrix are applied to construct a monotonic TG model for capturing discontinuities. To avoid making the scheme overly diffusive, we apply a flux-corrected transport (FCT) technique of Boris and Book to overcome the difficulty with anti-diffusive flux. In smooth flow regions, our strategyof developing the temporal and spatial high-order TG finite-element model is based on modified equation analysis. In regions where discontinuity is encountered, we resort to two dispersively more accurate models to make the prediction accuracy as high as that obtained in smooth cases. These models are developed using the entropy-increasing principle and the theory of group velocity. Guided by this theory, a slower group velocity should be used ahead of the shock. To avoid a train of post-shocks, free parameters should be chosen properly to obtain a group velocity which takes on a larger value than the exact phase velocity. In this paper, we also apply the entropy-increasing principle to determine free parameters introduced in the finite-element model. Under the entropy-increasing requirement, it is mandatory that coefficients of the even and odd derivative terms shown in the modified equation should change signs alternatively in order to avoid non-physical wiggles. Several benchmark problems have been investigated to confirm the integrity of these proposed characteristic models.