高效的特征值边界法:重温 CFL 条件

IF 7.2 2区 物理与天体物理 Q1 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
F.X. Trias , X. Álvarez-Farré , A. Alsalti-Baldellou , A. Gorobets , A. Oliva
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引用次数: 0

摘要

湍流的直接模拟和大涡流模拟通常采用显式时间方案求解。然而,这需要非常小的时间步长,因为按时间步长重新缩放的(线性化)动力系统特征值必须位于稳定区域内。因此,在实践中需要快速准确地估计离散对流项和扩散项的谱半径。这几乎总是通过所谓的 CFL 条件来实现。另一方面,现代超级计算系统的巨大异质性和复杂性阻碍了 CFD 代码跨平台的高效移植。在这方面,我们的主旨是:依靠一组最小的(代数)内核对于代码的可移植性和维护至关重要!在此背景下,这项工作的重点是计算上述对流矩阵和扩散矩阵的特征边界,这是确定 CFL 时间步长所必需的。为此,我们提出并测试了一种无需重新构建这些随时间变化的矩阵的廉价新方法。它只依赖于稀疏矩阵矢量乘积,其中只有矢量随时间变化。因此,无论是在现有代码中实现还是跨平台移植都很简单。在结构化笛卡尔网格和非结构化网格的不同测试案例中,证明了该方法的有效性和鲁棒性。最后,该方法与自适应时间方案相结合,与其他更传统的基于 CFL 的方法相比,大大提高了时间步长。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
An efficient eigenvalue bounding method: CFL condition revisited

Direct and large-eddy simulations of turbulence are often solved using explicit temporal schemes. However, this imposes very small time-steps because the eigenvalues of the (linearized) dynamical system, re-scaled by the time-step, must lie inside the stability region. In practice, fast and accurate estimations of the spectral radii of both the discrete convective and diffusive terms are therefore needed. This is virtually always done using the so-called CFL condition. On the other hand, the large heterogeneity and complexity of modern supercomputing systems are nowadays hindering the efficient cross-platform portability of CFD codes. In this regard, our leitmotiv reads: relying on a minimal set of (algebraic) kernels is crucial for code portability and maintenance! In this context, this work focuses on the computation of eigenbounds for the above-mentioned convective and diffusive matrices which are needed to determine the time-step à la CFL. To do so, a new inexpensive method, that does not require to re-construct these time-dependent matrices, is proposed and tested. It just relies on a sparse-matrix vector product where only vectors change on time. Hence, both implementation in existing codes and cross-platform portability are straightforward. The effectiveness and robustness of the method are demonstrated for different test cases on both structured Cartesian and unstructured meshes. Finally, the method is combined with a self-adaptive temporal scheme, leading to significantly larger time-steps compared with other more conventional CFL-based approaches.

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来源期刊
Computer Physics Communications
Computer Physics Communications 物理-计算机:跨学科应用
CiteScore
12.10
自引率
3.20%
发文量
287
审稿时长
5.3 months
期刊介绍: The focus of CPC is on contemporary computational methods and techniques and their implementation, the effectiveness of which will normally be evidenced by the author(s) within the context of a substantive problem in physics. Within this setting CPC publishes two types of paper. Computer Programs in Physics (CPiP) These papers describe significant computer programs to be archived in the CPC Program Library which is held in the Mendeley Data repository. The submitted software must be covered by an approved open source licence. Papers and associated computer programs that address a problem of contemporary interest in physics that cannot be solved by current software are particularly encouraged. Computational Physics Papers (CP) These are research papers in, but are not limited to, the following themes across computational physics and related disciplines. mathematical and numerical methods and algorithms; computational models including those associated with the design, control and analysis of experiments; and algebraic computation. Each will normally include software implementation and performance details. The software implementation should, ideally, be available via GitHub, Zenodo or an institutional repository.In addition, research papers on the impact of advanced computer architecture and special purpose computers on computing in the physical sciences and software topics related to, and of importance in, the physical sciences may be considered.
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