Enhanced spectral viscosity approximations for conservation laws
A spectral method for conservation laws is presented, which is based on the spectral viscosity (SV) method formerly introduced by \textit{E. Tadmor} [SIAM J. Numer. Anal. 26, No. 1, 30-44 (1989; Zbl 0667.65079)] and later developed with co-workers. The novelty here is that an enhanced edge detection method is used for locating shocks and contact discontinuities. That method relies on so-called concentration factors, and was previously introduced in a general context by \textit{A. Gelb} and \textit{G. Tadmor} [Appl. Comput. Harmon. Anal. 7, No. 1, 101-135 (1999; Zbl 0952.42001) and SIAM J. Numer. Anal. 38, No. 4, 1389-1408 (2000; Zbl 0990.42025)]. It appears to be well suited to identify the edges of discontinuities arising in approximate solutions of conservation laws computed by the SV-method. This enables the authors to profitably use the Gegenbauer post-processing method in order to cope with the Gibbs phenomenon [on this topic, see \textit{D. Gottlied} and \textit{C.-W. Shu}, SIAM Rev. 39, No. 4, 644-668 (1997; Zbl 0885.42003)]. Convincing numerical examples are provided for the inviscid Burgers equation with periodic boundary conditions and the Euler equations for a polytropic gas with Riemann data. It is to be noted that both the enhanced edge detection and the post-processing are performed only at the final time step, which minimizes the additional cost. An interesting feature of the results presented is that computed contact discontinuities are sharp and free of oscillations. Surprisingly though, there are spikes at the tips of rarefactions. The authors hope to improve this by optimizing the parameters in the Gegenbauer post-processing method.
scientific article; zbMATH DE number 1441149