混合隐式-显式通量重构方法
Hybridized Implicit-Explicit Flux Reconstruction Methods
AI总结：
针对工业规模湍流问题的计算成本瓶颈，提出结合混合化与标准通量重构的隐式-显式格式，通过传输条件实现整体守恒，数值实验验证其相比标准方法可实现数倍至十余倍的计算加速。
摘要：
对于工业规模的湍流问题，由于基础守恒律的显式时间离散化存在稳定性约束，计算成本可能会高得难以承受。另一方面，隐式方法允许使用更大的时间步长，但需要极高的计算资源。隐式-显式（IMEX，即Implicit-Explicit）公式结合了这两种时间推进方法，在域的非刚性部分使用显式方法，在刚性部分使用隐式方法。虽然已证明这类方法比典型的显式离散化快几个数量级，但它们的成本仍受限于隐式离散化部分。混合化将这些系统的求解规模降低到有效低维，与标准隐式方法相比，能以显著的加速比求解系统。本研究提出了一种结合混合化与标准通量重构（FR，即Flux Reconstruction）方法的IMEX格式，用于解决几何诱导的刚性问题。通过使用所谓的传输条件，将显式FR方法与混合隐式FR方法结合后，可得到整体守恒的公式。我们对一系列数值算例验证并应用了该方法，包括雷诺数为170万的多段翼型算例。结果表明，针对同一问题，该方法相比标准IMEX公式加速比为4倍，相比标准显式公式加速比至少为15倍。
英文摘要：
For turbulent problems of industrial scale, computational cost may become prohibitive due to the stability constraints associated with explicit time discretization of the underlying conservation laws. On the other hand, implicit methods allow for larger time-step sizes but require exorbitant computational resources. Implicit-explicit (IMEX) formulations combine both temporal approaches, using an explicit method in nonstiff portions of the domain and implicit in stiff portions. While these methods can be shown to be orders of magnitude faster than typical explicit discretizations, they are still limited by their implicit discretization in terms of cost. Hybridization reduces the scaling of these systems to an effective lower dimension, which allows the system to be solved at significant speedup factors compared to standard implicit methods. This work proposes an IMEX scheme that combines hybridized and standard flux reconstriction (FR) methods to tackle geometry-induced stiffness. By using the so-called transmission conditions, an overall conservative formulation can be obtained after combining both explicit FR and hybridized implicit FR methods. We verify and apply our approach to a series of numerical examples, including a multi-element airfoil at Reynolds number 1.7 million. Results demonstrate speedup factors of four against standard IMEX formulations and at least 15 against standard explicit formulations for the same problem.
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