The linear stability analysis on the WENO5 spatial discretization for solving the one-dimensional linear advection equation, combined with various fifth-order multistep methods, was presented in Motamed et al. (J Sci Comput 47(2):127–149, 2011). The purpose of this work is to further investigate the mechanism of oscillations observed in these time integrators when simulating shock front propagation. In particular, extrapolated backward differentiation formula (eBDF5), explicit Adams methed (Adams5) and a predictor–corrector method (PC5) are selected for detailed performance comparison. We first analyze how the non-convex combinations involved in these multistep methods restrict the time step-size and lead to possible pointwise oscillations. Subsequently, the nonlinear weights in the WENO5 scheme are used as indicators to capture the evolution of discontinuities with time and determine the stability of the multistep methods. Numerical results are also provided to confirm the analysis and review the qualifications of these multistep methods for shock front tracking.

Motamed等人提出了WENO5空间离散化的线性稳定性分析，用于求解一维线性对流方程，并结合了多种五阶多步法。（J Sci Comput 47（2）：127–149，2011）。这项工作的目的是进一步研究在模拟冲击波前传播时在这些时间积分器中观察到的振荡机理。特别是，选择了外推后向差分公式（eBDF5），显式Adams方法（Adams5）和预测器-校正器方法（PC5）进行详细的性能比较。我们首先分析这些多步方法中涉及的非凸组合如何限制时间步长并导致可能的逐点振荡。后来，WENO5方案中的非线性权重用作指示符，以捕获不连续性随时间的演变并确定多步法的稳定性。还提供了数值结果，以确认分析结果并审查这些多步方法在冲击前跟踪中的资格。