Stability of Linear Multistep Time Iterations with the WENO5 Discretization at Discontinuities
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Stability of Linear Multistep Time Iterations with the WENO5 Discretization at Discontinuities Jianying Zhang1 Received: 14 March 2020 / Revised: 6 July 2020 / Accepted: 18 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The linear stability analysis on the WENO5 spatial discretization for solving the onedimensional 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. Keywords Weighted essentially non-oscillatory scheme · Multistep method · Total variation diminishing · Runge–Kutta method · Shock propagation
1 Introduction High order weighted essentially non-oscillatory (WENO) schemes [4,8,13] are a popular class of spatial numerical discretization methods for solving hyperbolic conservation laws, as well as convection dominated partial differential equations arising from a wide variety of applied disciplines. The major advantage of WENO schemes is their non-oscillatory feature in capturing sharp discontinuity transitions while maintaining arbitrarily high order accuracy in smooth regions. Their robustness in computation is especially desirable when high resolution of delicate solution structures is necessary, such as simulating vortices and acoustic waves in complex fluid flows and pattern formation in developmental biology, etc. A detailed survey of related applications can be found in [14].
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Jianying Zhang [email protected] Department of Mathematics, Western Washington University, Bellingham, WA 98225, USA 0123456789().: V,-vol
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To accomplish the time integration of these equations, the WENO discretization is usually combined with the method of lines so that the spatially discretized PDEs of interest are converted to a system of ODEs, which can be solved via effective time discretization techniques. The most popular ones include total variation diminishing (TVD) Runge–Kutta methods, or termed in the literature, the strong stability preserving (SSP) methods [2,3,7,11,12,15]. Alternatively, time discretization based on the Lax–Wendroff procedure is also an option [9,16]. The linear stab
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