High-order residual distribution scheme for the time-dependent Euler equations of fluid dynamics
High-order residual distribution scheme for the time-dependent Euler equations of fluid dynamics
复制标题
DOI：
10.1016/j.camwa.2018.05.009
复制
发表时间：
2019-07-15
影响因子：
2.9
通讯作者：
Tokareva, Svetlana
中科院分区：
文献类型：
--
作者：
Abgrall, Remi;Bacigaluppi, Paola;Tokareva, Svetlana
本文在气体动力学的多维可压缩欧拉方程框架下，设计了一种高阶有限元形式的余量分布格式。该近似的优点在于利用连续有限元型近似技术对模型方程进行一般的空间离散化，同时避免了求解具有稀疏质量矩阵的大型线性系统的方法，而在经典有限元方法中，任何标准的ODE求解器都会伴随着这种方法来及时推进求解。在这项工作中，我们提出了一种新的剩余分布(RD)格式，它提供了欧拉方程在空间和时间上的光滑解的任意显式高阶近似。通过RD公式(Ricchiuto和Abgrall，2010；Abgrall，2006)与延迟校正(DEC)型方法(Liu等人，2008；Minion，2003)的耦合，方案的设计允许与需要求逆的及时更新相关联的矩阵是对角线的。使用Bernstein多项式作为形函数，确保了这个对角矩阵是可逆的，并确保了所得到的对角矩阵系数的严格正性。本工作是Abgrall等人的推广。(2016)和Abgrall(2017)到多维系统。我们对一维和二维Euler方程的几个具有挑战性的基准问题评估了我们的方法，该格式被证明是健壮的，并且在光滑解上达到了理论预测的高精度。(C)2018爱思唯尔有限公司。保留所有权利。
In the present work, a high order finite element type residual distribution scheme is designed in the framework of multidimensional compressible Euler equations of gas dynamics. The strengths of the proposed approximation rely on the generic spatial discretization of the model equations using a continuous finite element type approximation technique, while avoiding the solution of a large linear system with a sparse mass matrix which would come along with any standard ODE solver in a classical finite element approach to advance the solution in time. In this work, we propose a new Residual Distribution (RD) scheme, which provides an arbitrary explicit high order approximation of the smooth solutions of the Euler equations both in space and time. The design of the scheme via the coupling of the RD formulation (Ricchiuto and Abgrall, 2010; Abgrall, 2006) with a Deferred Correction (DeC) type method (Liu et al., 2008; Minion, 2003), allows to have the matrix associated to the update in time, which needs to be inverted, to be diagonal. The use of Bernstein polynomials as shape functions, guarantees that this diagonal matrix is invertible and ensures strict positivity of the resulting diagonal matrix coefficients. This work is the extension of Abgrall et al. (2016) and Abgrall (2017) to multidimensional systems. We have assessed our method on several challenging benchmark problems for one- and two-dimensional Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions. (C) 2018 Elsevier Ltd. All rights reserved.