Abstract
We present a new version of conservative ADER-WENO finite volume schemes, in which both the high order spatial reconstruction as well as the time evolution of the reconstruction polynomials in the local space-time predictor stage are performed in primitive variables, rather than in conserved ones. To obtain a conservative method, the underlying finite volume scheme is still written in terms of the cell averages of the conserved quantities. Therefore, our new approach performs the spatial WENO reconstruction twice: the first WENO reconstruction is carried out on the known cell averages of the conservative variables. The WENO polynomials are then used at the cell centers to compute point values of the conserved variables, which are subsequently converted into point values of the primitive variables. This is the only place where the conversion from conservative to primitive variables is needed in the new scheme. Then, a second WENO reconstruction is performed on the point values of the primitive variables to obtain piecewise high order reconstruction polynomials of the primitive variables. The reconstruction polynomials are subsequently evolved in time with a novel space-time finite element predictor that is directly applied to the governing PDE written in primitive form. The resulting space-time polynomials of the primitive variables can then be directly used as input for the numerical fluxes at the cell boundaries in the underlying conservative finite volume scheme. Hence, the number of necessary conversions from the conserved to the primitive variables is reduced to just one single conversion at each cell center. We have verified the validity of the new approach over a wide range of hyperbolic systems, including the classical Euler equations of gas dynamics, the special relativistic hydrodynamics (RHD) and ideal magnetohydrodynamics (RMHD) equations, as well as the Baer-Nunziato model for compressible two-phase flows. In all cases we have noticed that the new ADER schemes provide less oscillatory solutions when compared to ADER finite volume schemes based on the reconstruction in conserved variables, especially for the RMHD and the Baer-Nunziato equations. For the RHD and RMHD equations, the overall accuracy is improved and the CPU time is reduced by about 25 %. Because of its increased accuracy and due to the reduced computational cost, we recommend to use this version of ADER as the standard one in the relativistic framework. At the end of the paper, the new approach has also been extended to ADER-DG schemes on space-time adaptive grids (AMR).
Highlights
Since their introduction by Toro and Titarev (Toro et al ; Titarev and Toro ; Toro and Titarev ; Titarev and Toro ; Toro and Titarev ), ADER schemes for hyperbolic partial differential equations (PDE) have been improved and developed along different directions
ADER schemes have been used in combination with discontinuous Galerkin methods, even in the presence of shock waves and other discontinuities within the flow, thanks to a novel a posteriori sub-cell finite volume limiter technique based on the MOOD approach (Clain et al ; Diot et al ), that is designed to stabilize the discrete solution wherever the DG approach fails and produces spurious oscillations or negative densities and pressures (Dumbser et al ; Zanotti et al a; Zanotti et al b)
In the following we explore the properties of the new ADER-WENO finite volume scheme by solving a wide set of test problems belonging to four different systems of equations: the classical Euler equations, the relativistic hydrodynamics (RHD) and magnetohydrodynamics (RMHD) equations and the Baer-Nunziato equations for compressible two-phase flows
Summary
Since their introduction by Toro and Titarev (Toro et al ; Titarev and Toro ; Toro and Titarev ; Titarev and Toro ; Toro and Titarev ), ADER (arbitrary high order derivatives) schemes for hyperbolic partial differential equations (PDE) have been improved and developed along different directions. A conversion from the conserved to the primitive variables is necessary for the calculations of the fluxes, and this operation, which is never analytic for such systems of equations, is rather expensive For these reasons it would be very desirable to have an ADER scheme in which both the reconstruction and the subsequent local space-time discontinuous Galerkin predictor are performed in primitive variables. This marks a striking difference with respect to traditional ADER schemes, in which such polynomials are instead computed in conserved variables and are denoted as qh (see, e.g. Hidalgo and Dumbser ). The path in Eqs. ( ) and ( ) is the same segment path adopted in ( ) for the computation of the jumps Di
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