2015/11/15 by Olindo Zanotti, Michael Dumbser, Zanotti, Olindo +1
Earth and Planetary Sciences · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Geophysics and Gravity Measurements #High Energy Astrophysical Phenomena (astro-ph.HE) #Ionosphere and magnetosphere dynamics #Meteorological Phenomena and Simulations #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1511.04728
openalex publication_date 2015/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We present a new version of conservative ADER-WENO finite volume schemes, in\nwhich both the high order spatial reconstruction as well as the time evolution\nof the reconstruction polynomials in the local space-time predictor stage are\nperformed in primitive variables, rather than in conserved ones. Since the\nunderlying finite volume scheme is still written in terms of cell averages of\nthe conserved quantities, our new approach performs the spatial WENO\nreconstruction twice: the first WENO reconstruction is carried out on the known\ncell averages of the conservative variables. The WENO polynomials are then used\nat the cell centers to compute point values of the conserved variables, which\nare converted into point values of the primitive variables. A second WENO\nreconstruction is performed on the point values of the primitive variables to\nobtain piecewise high order reconstruction polynomials of the primitive\nvariables. The reconstruction polynomials are subsequently evolved in time with\na novel space-time finite element predictor that is directly applied to the\ngoverning PDE written in primitive form. We have verified the validity of the\nnew approach over the classical Euler equations of gas dynamics, the special\nrelativistic hydrodynamics (RHD) and ideal magnetohydrodynamics (RMHD)\nequations, as well as the Baer-Nunziato model for compressible two-phase flows.\nIn all cases we have noticed that the new ADER schemes provide less oscillatory\nsolutions when compared to ADER finite volume schemes based on the\nreconstruction in conserved variables, especially for the RMHD and the\nBaer-Nunziato equations. For the RHD and RMHD equations, the accuracy is\nimproved and the CPU time is reduced by about 25%. We recommend to use this\nversion of ADER as the standard one in the relativistic framework. The new\napproach can be extended to ADER-DG schemes on space-time adaptive grids.\n