Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems
Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems
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DOI：
10.1016/j.jcp.2006.06.043
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发表时间：
2007-02
期刊：
影响因子：
--
通讯作者：
M. Dumbser;M. Käser
中科院分区：
文献类型：
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作者：
M. Dumbser;M. Käser
在本文中，我们提出了一个无振荡的有限体积格式的任意精度的空间和时间的非结构网格上的二维和三维空间的ADER方法求解线性双曲型方程组。关键点是一个新的重建算子，利用原来开发的技术在不连续伽辽金有限元框架。首先，我们使用层次正交基进行重建。其次，重建不是在物理坐标中进行的，而是在参考坐标系中进行的，这消除了缩放效应，从而避免了病态重建矩阵。为了实现非振荡特性，我们提出了一种新的韦诺重建技术，不重建点值，但整个多项式，可以很容易地评估和区分在任何点。我们表明，由于特殊的重建韦诺振荡指标可以计算在一个网格独立的方式由一个简单的二次功能。我们的韦诺计划不遭受负权重的问题，如先前在文献中所描述的，因为线性权重不用于提高精度。准确性是通过仅仅把一个大的线性重量的中心模板。由此产生的一步ADER有限体积格式，以这种方式获得的每个元素和时间步长只执行一个非线性韦诺重建，因此可以非常有效地实现，即使在三维空间的非结构化网格。我们表明，所提出的方法在空间和时间上的非结构化三角形和四面体网格在两个和三个空间维度，分别达到六阶的收敛结果。
In this article we present a non-oscillatory finite volume scheme of arbitrary accuracy in space and time for solving linear hyperbolic systems on unstructured grids in two and three space dimensions using the ADER approach. The key point is a new reconstruction operator that makes use of techniques developed originally in the discontinuous Galerkin finite element framework. First, we use a hierarchical orthogonal basis to perform reconstruction. Second, reconstruction is not done in physical coordinates, but in a reference coordinate system which eliminates scaling effects and thus avoids ill-conditioned reconstruction matrices. In order to achieve non-oscillatory properties, we propose a new WENO reconstruction technique that does not reconstruct point-values but entire polynomials which can easily be evaluated and differentiated at any point. We show that due to the special reconstruction the WENO oscillation indicator can be computed in a mesh-independent manner by a simple quadratic functional. Our WENO scheme does not suffer from the problem of negative weights as previously described in the literature, since the linear weights are not used to increase accuracy. Accuracy is obtained by merely putting a large linear weight on the central stencil. The resulting one-step ADER finite volume scheme obtained in this way performs only one nonlinear WENO reconstruction per element and time step and thus can be implemented very efficiently even for unstructured grids in three space dimensions. We show convergence results obtained with the proposed method up to sixth order in space and time on unstructured triangular and tetrahedral grids in two and three space dimensions, respectively.