Optimal embedded pair Runge-Kutta schemes for pseudo-time stepping
Optimal embedded pair Runge-Kutta schemes for pseudo-time stepping
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用于伪时间步进的最佳嵌入对 Runge-Kutta 方案
DOI：
10.1016/j.jcp.2020.109499
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
P. Vincent
中科院分区：
文献类型：
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作者：
Brian C. Vermeire;Niki A. Loppi;P. Vincent
当使用人工可压缩性方法（ACM）时，在每个物理时间步长求解稳态伪时间问题以强制不可压缩性[1]。由于现代硬件架构的性质，使用显式Runge-Kutta（RK）方法解决这个伪时间问题是一种有吸引力的方法[2]。显式RK方法还可以与收敛加速技术结合，包括局部自适应伪时间步进[3]，其基于局部误差估计动态调整每个自由度的伪时间步长Δτ，该局部误差估计充当局部稳定性的代理。获得局部误差估计的一种方法是使用嵌入的显式RK方法对。这两种方案之间的差异产生的误差估计，然后可以用来最大化的伪时间步长，同时，避免违反稳定性限制。这使得伪时间步长的选择自动化，并且允许其基于变化的稳定性约束而改变。此外，多项式多重网格（P-多重网格）可用于在使用高阶空间离散时加速收敛，例如不连续Galerkin（DG）和通量重建（FR）方法[4]。以前已经证明，可以开发最佳显式Runge-Kutta格式以最大化给定空间离散的允许时间步长，使用Ketcheson和Ahmidia介绍的技术[5]。这已被用于为谱差分方法[6]生成最佳稳定性多项式，以及为不可压缩流[7]使用伪时间步进的通量重建方法生成最佳稳定性多项式。在目前的研究中，我们扩展了Vermeire等人的工作。[7]一个嵌入的最优显式RK方法对，它提供误差估计，以确定Δτ何时接近稳定性极限。传统的嵌入对方案，如众所周知的RK 45和RK 34方法，由两个方案组成，具有相同的级数和内部级系数，但精度不同。在每个步骤结束时计算时间误差估计，并且调整时间步长以将该误差保持在阈值以下。然而，在伪时间步进的上下文中，我们感兴趣的是最大化Δτ，而不是保持精度。我们提出了一族新的嵌入对RK方法来加速伪时间问题的收敛。这些是新颖的，因为两个成员都是一阶精度的，它们是一致的，同时具有不同的级数而不是精度的阶数，并且两者都可以针对特定的空间离散化进行优化以最大化Δτ。通过设计，级数较少的格式的稳定极限略低于级数较多的格式。因此，当Δτ接近具有较少级数的方案的稳定性极限时，两个方案之间的误差迅速增加，这可以用作Δτ何时接近具有较多级数的方案的稳定性极限的指示。在这个简短的技术说明中，我们概述了这些嵌入对计划的建设，他们的优化为一个给定的空间离散化，并证明他们的效用相对于现有的显式计划的非定常不可压缩流。
When using the Artificial Compressibility Method (ACM) a steady-state pseudo-time problem is solved each physical time step to enforce incompressibility [1]. Due to the nature of modern hardware architectures, solving this pseudo-time problem using explicit Runge-Kutta (RK) methods is an appealing approach [2]. Explicit RK methods can also be combined with convergence acceleration techniques including locally adaptive pseudo-time stepping [3], which dynamically adjusts the pseudo-time step size Δτ of each degree of freedom based on local error estimates that act as a proxy for local stability. One approach for obtaining a local error estimate is with an embedded pair of explicit RK methods. The difference between these two schemes yields an error estimate that can then be used to maximize the pseudo-time step size and, simultaneously, avoid violating the stability limit. This automates the choice of pseudo-time step size, and allows it to change based on varying stability constraints. Furthermore, polynomial multigrid (P-multigrid) can be used to accelerate convergence when using high-order spatial discretizations, such as the Discontinuous Galerkin (DG) and Flux Reconstruction (FR) approaches [4].It has previously been demonstrated that optimal explicit Runge-Kutta schemes can be developed to maximize the allowable time step size for a given spatial discretization, using the technique introduced by Ketcheson and Ahmidia [5]. This has been used to generate optimal stability polynomials for spectral difference methods [6], and for flux reconstruction methods using pseudo-time stepping for incompressible flows [7]. In the current study, we extend the work of Vermeire et al.[7] to an embedded pair of optimal explicit RK methods, which provide error estimates to determine when Δτ is near the stability limit. Conventional embedded pair schemes, such as the well-known RK45 and RK34 methods, consist of two schemes with the same number of stages and internal stage coefficients, but different orders of accuracy. A temporal error estimate is computed at the end of each step and the time step size is adjusted to maintain this error below a threshold. However, in the context of pseudo-time stepping, we are interested in maximizing Δτ, rather than maintaining accuracy. We propose a new family of embedded pair RK methods to accelerate convergence of the pseudo-time problem. These are novel in that both members are first-order accurate, they are consistent while having different numbers of stages rather than orders of accuracy, and both can be optimized for a particular spatial discretization to maximize Δτ. By design, the stability limit of the scheme with fewer stages is slightly lower than that of the scheme with more stages. Hence, the error between the two schemes increases rapidly when Δτ is near the stability limit of the scheme with fewer stages, which can be used as an indicator for when Δτ is near the stability limit of the scheme with more stages. In this short technical note we outline the construction of these embedded pair schemes, their optimization for a given spatial discretization, and demonstrate their utility relative to existing explicit schemes for unsteady incompressible flows.