Stability of Linear Multistep Time Iterations with the WENO5 Discretization at Discontinuities.

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. [ABSTRACT FROM AUTHOR]