High-order unstructured methods have become a popular choice for the simulation of complex unsteady flows. Flux reconstruction (FR) is a high-order spatial discretization method, which has been found to be particularly accurate for scale-resolving simulations of complex phenomena. In addition, it has been shown to provide sufficient dissipation for implicit large-eddy simulation (ILES). In conjunction with an FR discretization, an appropriate temporal scheme must be chosen. A common choice is explicit schemes due to their efficiency and ease of implementation. However, these methods usually require a small time-step size to remain stable. Recently, the development of optimal explicit Runge–Kutta (OERK) schemes has enabled stable simulations with larger time-step sizes. Hence, we analyze the fully-discrete properties of the FR method with OERK temporal schemes. We show results for first, second, third, fourth and eighth-order OERK schemes. We observe that OERK schemes modify the spectral behaviour of the semidiscretization. In particular, dissipation decreases in the region of high wavenumbers. We observe that higher-order OERK schemes require a smaller time step than the low-order schemes. However, they follow the dispersion relations of the FR scheme for a larger range of wavenumbers. We validate our analysis with simple advection test cases. It was observed that first and second-degree temporal schemes introduce a relatively large amount of error in the solutions. A one-dimensional ILES test case showed that, as long as the time-step size is not in the vicinity of the stability limit, results are generally similar to classical RK schemes.

高阶非结构化方法已成为模拟复杂非恒定流的流行选择。磁通重构（FR）是一种高阶空间离散化方法，已发现对于复杂现象的尺度解析模拟特别准确。此外，它已显示出为隐式大涡模拟（ILES）提供足够的耗散。结合FR离散化，必须选择适当的时间方案。常见的选择是显式方案，因为它们的效率高且易于实施。但是，这些方法通常需要较小的时间步长才能保持稳定。最近，最佳显式Runge-Kutta（OERK）方案的开发已经实现了具有较大时间步长的稳定仿真。因此，我们使用OERK时间方案分析FR方法的完全离散性质。我们显示了一阶，二阶，三阶，四阶和八阶OERK方案的结果。我们观察到OERK方案修改了半离散化的频谱行为。特别地，在高波数的区域中耗散减小。我们观察到，高阶OERK方案比低阶方案需要更小的时间步长。但是，对于较大的波数范围，它们遵循FR方案的色散关系。我们使用简单的对流测试案例来验证我们的分析。已经观察到，一阶和二阶时间方案在解中引入了相对大量的误差。一维ILES测试案例表明，只要时间步长不在稳定性极限附近，