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Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables.
Computational Astrophysics and Cosmology Pub Date : 2016-01-13 , DOI: 10.1186/s40668-015-0014-x
Olindo Zanotti 1 , Michael Dumbser 1
Affiliation  

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).

中文翻译:

基于WENO重构和原始变量时空预测器的高效保守ADER方案。

我们提出了一种新的保守ADER-WENO有限体积方案,其中高阶空间重构以及局部时空预测器阶段中重构多项式的时间演化都是在原始变量中执行,而不是在守恒变量中执行那些。为了获得保守的方法,仍将根据守恒量的像元平均值来编写基本的有限体积方案。因此,我们的新方法执行了两次空间WENO重建:第一次WENO重建是在保守变量的已知单元平均上执行的。然后,在单元中心使用WENO多项式来计算保守变量的点值,然后将其转换为原始变量的点值。这是新方案中唯一需要从保守变量转换为原始变量的地方。然后,对原始变量的点值进行第二次WENO重构,以获得分段的原始变量的高阶重构多项式。随后使用新颖的时空有限元预测器在时间上演化重构多项式,该预测器直接应用于以原始形式编写的控制PDE。然后,可以将原始变量的时空多项式直接用作基础保守有限体积方案中单元边界处数字通量的输入。因此,从保守变量到原始变量的必要转换次数减少为每个单元中心的一次转换。我们已经验证了该新方法在广泛的双曲系统中的有效性,包括经典的气体动力学欧拉方程,特殊的相对论流体动力学(RHD)和理想磁流体动力学(RMHD)方程以及Baer-Nunziato模型可压缩的两相流。在所有情况下,我们都注意到,与基于守恒变量重构的ADER有限体积方案相比,新的ADER方案提供的振荡解更少,尤其是对于RMHD和Baer-Nunziato方程而言。对于RHD和RMHD公式,整体精度得到了改善,CPU时间减少了约25%。由于提高了准确性,并且降低了计算成本,因此我们建议在相对论框架中使用此版本的ADER作为标准版本。在本文的最后,
更新日期:2016-01-13
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