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A New Mapped Weighted Essentially Non-oscillatory Scheme

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Abstract

The weighted essentially non-oscillatory (WENO) methods are a popular high-order spatial discretization for hyperbolic partial differential equations. Recently Henrick et al. (J. Comput. Phys. 207:542–567, 2005) noted that the fifth-order WENO method by Jiang and Shu (J. Comput. Phys. 126:202–228, 1996) is only third-order accurate near critical points of the smooth regions in general. Using a simple mapping function to the original weights in Jiang and Shu (J. Comput. Phys. 126:202–228, 1996), Henrick et al. developed a mapped WENO method to achieve the optimal order of accuracy near critical points. In this paper we study the mapped WENO scheme and find that, when it is used for solving the problems with discontinuities, the mapping function in Henrick et al. (J. Comput. Phys. 207:542–567, 2005) may amplify the effect from the non-smooth stencils and thus cause a potential loss of accuracy near discontinuities. This effect may be difficult to be observed for the fifth-order WENO method unless a long time simulation is desired. However, if the mapping function is applied to seventh-order WENO methods (Balsara and Shu in J. Comput. Phys. 160:405–452, 2000), the error can increase much faster so that it can be observed with a moderate output time. In this paper a new mapping function is proposed to overcome this potential loss of accuracy.

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Authors and Affiliations

  1. Faculty of Mathematics & Statistics, Wuhan University, Wuhan, Hubei, 430072, P.R. China

    Hui Feng, Fuxing Hu & Rong Wang

Authors
  1. Hui Feng
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  2. Fuxing Hu
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  3. Rong Wang
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Corresponding author

Correspondence to Rong Wang.

Additional information

The work of H. Feng and R. Wang was par tially supported by National Natural Science Foundation of China (No. 10971159) and NCET of China (No. NCET-06-0614).

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Feng, H., Hu, F. & Wang, R. A New Mapped Weighted Essentially Non-oscillatory Scheme. J Sci Comput 51, 449–473 (2012). https://doi.org/10.1007/s10915-011-9518-y

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  • DOI: https://doi.org/10.1007/s10915-011-9518-y

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