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A Spectral Study on the Dissipation and Dispersion of the WENO Schemes

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The dissipation and dispersion (spectral) properties of the nonlinear fifth order classical weighted essentially non-oscillatory finite difference scheme (WENO-JS5) and its improved version (WENO-Z5) using the approximate dispersion relation (ADR) (Pirozzoli in J Comput Phys 219:489–497, 2006) and the nonlinear spectral analysis (NSA) (Fauconnier et al. in J Comput Phys 228(6):1830–1861, 2009) are studied. Unlike the previous studies, the influences of the sensitivity parameter in the definition of the WENO nonlinear weights are also included for completeness. The fifth order upwinded central linear scheme (UW5) serves as the reference and benchmark for the purpose of comparison. The spectral properties of the WENO differentiation operator is well predicted theoretically by the ADR and validated numerically by the simulations of the WENO schemes in solving the scalar linear advection equation. In a long time simulation with an initial broadband wave, the WENO schemes generate spurious high modes with amplitude and spread of wavenumbers depend on the value of the sensitivity parameter. The NSA is applied to investigate the statistical nonlinear behavior, due to the nonlinear stencils adaptation of the WENO schemes, with a large set of initial conditions consisting of synthetic scalar fields with a prescribed energy spectrum and random phases. The statistics indicate that there is a small probability of an existence of a mild anti-dissipation in the low wavenumber range regardless of the size of the sensitivity parameter. Numerical examples demonstrate that the WENO-Z5 scheme is not only less dissipative and dispersive but also less sensitive to random phases than the WENO-JS5 scheme. Furthermore, a sensitivity parameter adaptive technique, in which its value depends on the local smoothness of the solution at a given spatial location and time, is introduced for solving a linear advection problem with a discontinuous initial condition. The preliminary result shows that the solution computed by the sensitivity parameter adaptive WENO-Z5 scheme agrees well with those computed by the WENO-Z5 scheme and the UW5 scheme in regions containing discontinuities and smooth solutions, respectively.

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Acknowledgments

The authors would like to express our sincere gratitude to Prof. Pirozzoli and Dr. Fauconnier for their valuable insights and comments during the course of this study. The authors would like to acknowledge the funding support of this research by China Postdoctoral Science Foundation (2012M521374, 2013T60684), Natural Science Foundation of Shandong Province (ZR2012AQ003), National Natural Science Foundation of China (11201441), Fundamental Research Funds for the Central Universities (201362033) and the startup funding by the Ocean University of China (Don). The authors (Don, Jia) would like to thank the School of Mathematical Sciences of Ocean University of China for hosting their visits in the First Summer Workshop in Advanced Research in Applied Mathematics and Scientific Computing 2013.

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Authors and Affiliations

  1. School of Aerospace Engineering, Beijing Institute of Technology, Beijing, China

    Feilin Jia

  2. School of Mathematical Sciences, Ocean University of China, Qingdao, China

    Zhen Gao & Wai Sun Don

  3. Division of Applied Mathematics, Brown University, Providence, RI,  20912, USA

    Wai Sun Don

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  1. Feilin Jia
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  2. Zhen Gao
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  3. Wai Sun Don
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Correspondence to Zhen Gao.

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Jia, F., Gao, Z. & Don, W.S. A Spectral Study on the Dissipation and Dispersion of the WENO Schemes. J Sci Comput 63, 49–77 (2015). https://doi.org/10.1007/s10915-014-9886-1

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  • DOI: https://doi.org/10.1007/s10915-014-9886-1

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