Volume 10, Issue 6
Improved Symmetry Property of High Order Weighted Essentially Non-Oscillatory Finite Difference Schemes for Hyperbolic Conservation Laws

Wai Sun Don, Peng Li, Kwun Ying Wong & Zhen Gao

Adv. Appl. Math. Mech., 10 (2018), pp. 1418-1439.

Published online: 2018-09

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  • Abstract

This study aims to investigate the rapid loss of numerical symmetry for problems with symmetrical initial conditions and boundary conditions when solved by the seventh and higher order nonlinear characteristic-wise weighted essentially non-oscillatory (WENO) finite difference schemes. Using the one-dimensional double rarefaction wave problem and the Sedov blast-wave problems, and the twodimensional Rayleigh-Taylor instability (RTI) problem as examples, we illustrate numerically that the sensitive interaction of the round-off error due to the numerical unstable explicit form of the local lower order smoothness indicators in the nonlinear weights definition, which are often given and used in the literature, and the nonlinearity of the WENO scheme are responsible for the rapid growth of asymmetry of an otherwise symmetric problem. An equivalent but compact and numerical stable compact form of the local lower order smoothness indicators is suggested for delaying the onset of and reducing the magnitude of the symmetry error. The benefits of using the compact form of the local lower order smoothness indicators should also be applicable to non-symmetrical strongly non-linear problems in terms of improved numerical stability, reduced rounding errors and increased computational efficiency

  • Keywords

Weighted essentially non-oscillatory symmetry smoothness indicator hyperbolic conservation laws.

  • AMS Subject Headings

65M06 65N06

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COPYRIGHT: © Global Science Press

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@Article{AAMM-10-1418, author = {}, title = {Improved Symmetry Property of High Order Weighted Essentially Non-Oscillatory Finite Difference Schemes for Hyperbolic Conservation Laws}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {6}, pages = {1418--1439}, abstract = {

This study aims to investigate the rapid loss of numerical symmetry for problems with symmetrical initial conditions and boundary conditions when solved by the seventh and higher order nonlinear characteristic-wise weighted essentially non-oscillatory (WENO) finite difference schemes. Using the one-dimensional double rarefaction wave problem and the Sedov blast-wave problems, and the twodimensional Rayleigh-Taylor instability (RTI) problem as examples, we illustrate numerically that the sensitive interaction of the round-off error due to the numerical unstable explicit form of the local lower order smoothness indicators in the nonlinear weights definition, which are often given and used in the literature, and the nonlinearity of the WENO scheme are responsible for the rapid growth of asymmetry of an otherwise symmetric problem. An equivalent but compact and numerical stable compact form of the local lower order smoothness indicators is suggested for delaying the onset of and reducing the magnitude of the symmetry error. The benefits of using the compact form of the local lower order smoothness indicators should also be applicable to non-symmetrical strongly non-linear problems in terms of improved numerical stability, reduced rounding errors and increased computational efficiency

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2017-0292}, url = {http://global-sci.org/intro/article_detail/aamm/12717.html} }
TY - JOUR T1 - Improved Symmetry Property of High Order Weighted Essentially Non-Oscillatory Finite Difference Schemes for Hyperbolic Conservation Laws JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1418 EP - 1439 PY - 2018 DA - 2018/09 SN - 10 DO - http://dor.org/10.4208/aamm.OA-2017-0292 UR - https://global-sci.org/intro/aamm/12717.html KW - Weighted essentially non-oscillatory KW - symmetry KW - smoothness indicator KW - hyperbolic conservation laws. AB -

This study aims to investigate the rapid loss of numerical symmetry for problems with symmetrical initial conditions and boundary conditions when solved by the seventh and higher order nonlinear characteristic-wise weighted essentially non-oscillatory (WENO) finite difference schemes. Using the one-dimensional double rarefaction wave problem and the Sedov blast-wave problems, and the twodimensional Rayleigh-Taylor instability (RTI) problem as examples, we illustrate numerically that the sensitive interaction of the round-off error due to the numerical unstable explicit form of the local lower order smoothness indicators in the nonlinear weights definition, which are often given and used in the literature, and the nonlinearity of the WENO scheme are responsible for the rapid growth of asymmetry of an otherwise symmetric problem. An equivalent but compact and numerical stable compact form of the local lower order smoothness indicators is suggested for delaying the onset of and reducing the magnitude of the symmetry error. The benefits of using the compact form of the local lower order smoothness indicators should also be applicable to non-symmetrical strongly non-linear problems in terms of improved numerical stability, reduced rounding errors and increased computational efficiency

Wai Sun Don, Peng Li, Kwun Ying Wong & Zhen Gao. (2020). Improved Symmetry Property of High Order Weighted Essentially Non-Oscillatory Finite Difference Schemes for Hyperbolic Conservation Laws. Advances in Applied Mathematics and Mechanics. 10 (6). 1418-1439. doi:10.4208/aamm.OA-2017-0292
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