Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables

We present a new version of conservative ADER-WENO finite volume schemes, in which both the high order spatialreconstruction as well as the time evolution of the reconstruction polynomials in the local space-time predictorstage are performed inprimitivevariables, rather than in conserved ones. To obtain a conservative method, theunderlying finite volume scheme is still written in terms of the cell averages of the conserved quantities. Therefore,our new approach performs the spatial WENO reconstructiontwice:thefirstWENO reconstruction is carried out onthe knowncell averagesof the conservative variables. The WENO polynomials are then used at the cell centers tocomputepoint valuesof theconserved variables, which are subsequently converted intopoint valuesof theprimitivevariables. This is the only place where the conversion from conservative to primitive variables is needed in the newscheme. Then, asecondWENO reconstruction is performed on the point values of the primitive variables to obtainpiecewise high order reconstruction polynomials of the primitive variables. The reconstruction polynomials aresubsequently evolved in time with anovelspace-time finite element predictor that is directly applied to thegoverning PDE written inprimitive form. The resulting space-time polynomials of the primitive variables can then bedirectly used as input for the numerical fluxes at the cell boundaries in the underlyingconservativefinite volumescheme. Hence, the number of necessary conversions from the conserved to the primitive variables is reduced tojustone single conversionat each cell center. We have verified the validity of the new approach over a wide range ofhyperbolic 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 compressibletwo-phase flows. In all cases we have noticed that the new ADER schemes provideless oscillatory solutionswhencompared to ADER finite volume schemes based on the reconstruction in conserved variables, especially for theRMHD and the Baer-Nunziato equations. For the RHD and RMHD equations, the overall accuracy is improved and theCPU 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).