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.

翻译：对于工业规模湍流问题，由于底层守恒定律显式时间离散的稳定性约束，计算成本可能变得过高。另一方面，隐式方法允许更大的时间步长，但需要过多的计算资源。隐式-显式（IMEX）方法结合了这两种时间推进方式，在非刚性区域使用显式方法，在刚性区域使用隐式方法。虽然这些方法比典型显式离散方法快数个量级，但其成本仍受限于隐式离散部分。混合化方法可将这些系统的规模降至有效低维，从而相比标准隐式方法实现显著加速比。本文提出一种结合混合化与标准通量重构（FR）方法的IMEX格式，用于解决几何诱导的刚性。通过使用所谓的传输条件，可在结合显式FR与混合化隐式FR方法后获得整体守恒的公式。我们通过一系列数值算例验证并应用该方法，包括雷诺数170万的多元件翼型。结果表明，对于相同问题，该方法相比标准IMEX格式实现四倍加速比，相比标准显式格式加速比至少达15倍。