A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes
A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes
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DOI：
10.1016/j.jcp.2008.05.025
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发表时间：
2008-09-10
影响因子：
4.1
通讯作者：
Munz, Claus-Dieter
中科院分区：
文献类型：
--
作者：
Dumbser, Michael;Balsara, Dinshaw S.;Munz, Claus-Dieter
本文将一种保守的最小二乘多项式重构算子应用于间断Galerkin方法。在第一种情况下，N次的分段多项式被用作测试函数以及表示在时间步开始时每个元素中的数据。然而，这些数据的时间演化和通量计算然后用不同的次数M >= N的分段多项式集合来完成，这些分段多项式集合是从N次的基础多项式重建的。这种方法产生一个一般的，统一的框架，包含作为两个特殊情况下，经典的高阶有限体积（FV）计划（N = 0），以及通常的间断Galerkin（DG）方法（N = M）。在第一种情况下，多项式是从细胞平均重建，对于后者，重建减少到单位算子。通过选择N不等于0且M > N，生成了一类全新的数值格式。重建算子适用于任意多项式次数N和M的二维和三维非结构三角形和四面体网格，为了提供与空间离散算子具有相同形式精度的高阶精度一步时间积分，重建的M次多项式数据在每个单元内使用一种新的局部连续时空Galerkin方法在时间上局部演化。作为这种方法的结果，我们得到，作为一个高阶准确的预测，空间-时间多项式的守恒变量的矢量和物理通量和源项，然后可以用在一个自然的方式来构建非常有效的全离散和正交自由的一步计划。这一特点是特别重要的DG计划在三个空间维，在那里的成本数值求积可能成为昂贵的非常高的精度orders.数值收敛研究的所有成员的新的一般类的建议计划显示高达六阶精度在空间和时间上的非结构化的二维和三维网格两个非常突出的非线性双曲型系统，即可压缩气体动力学的欧拉方程和理想磁流体动力学（MHD）方程。结果表明，新的中间格式（N不等于0，M > N）的时间推进方法在计算效率上优于传统的有限体积格式或DG格式.最后，在非结构网格上求解了大量的试验问题，其中所提出的新的时间推进方法被应用于理想和相对论MHD方程以及非线性弹性问题，使用标准的高阶韦诺有限体积空间离散化来科普间断解。(c)2008年爱思唯尔公司All rights reserved.
In this article, a conservative least-squares polynomial reconstruction operator is applied to the discontinuous Galerkin method. In a first instance, piecewise polynomials of degree N are used as test functions as well as to represent the data in each element at the beginning of a time step. The time evolution of these data and the flux computation, however,are then done with a different set of piecewise polynomials of degree M >= N, which are reconstructed from the underlying polynomials of degree N. This approach yields a general, unified framework that contains as two special cases classical high order finite volume (FV) schemes (N = 0) as well as the usual discontinuous Galerkin (DG) method (N = M). In the first case, the polynomial is reconstructed from cell averages, for the latter, the reconstruction reduces to the identity operator. A completely new class of numerical schemes is generated by choosing N not equal 0 and M > N. The reconstruction operator is implemented for arbitrary polynomial degrees N and M oil unstructured triangular and tetrahedral meshes in two and three space dimensions.To provide a high order accurate one-step time integration of the same formal order of accuracy as the spatial discretization operator, the (reconstructed) polynomial data of degree M are evolved in time locally inside each element using a new local continuous space-time Galerkin method. As a result of this approach, we obtain, as a high order accurate predictor, space-time polynomials for the vector of conserved variables and for the physical fluxes and source terms, which then can be used in a natural way to construct very efficient fully-discrete and quadrature-free one-step schemes. This feature is particularly important for DG schemes in three space dimensions, where the cost of numerical quadrature may become prohibitively expensive for very high orders of accuracy.Numerical convergence studies of all members of the new general class of proposed schemes are shown up to sixth-order of accuracy in space and time on unstructured two- and three-dimensional meshes for two very prominent nonlinear hyperbolic systems, namely for the Euler equations of compressible gas dynamics and the equations of ideal magnetohydrodynamics (MHD). The results indicate that the new class of intermediate schemes (N not equal 0, M > N) is computationally more efficient than classical finite volume or DG schemes.Finally, a large set of interesting test cases is solved on unstructured meshes, where the proposed new time stepping approach is applied to the equations of ideal and relativistic MHD as well as to nonlinear elasticity, using a standard high order WENO finite volume discretization in space to cope with discontinuous solutions. (c) 2008 Elsevier Inc. All rights reserved.