Improved Symmetry Property of High Order Weighted Essentially Non-Oscillatory Finite Difference Schemes for Hyperbolic Conservation Laws

Improved Symmetry Property of High Order Weighted Essentially Non-Oscillatory Finite Difference Schemes for Hyperbolic Conservation Laws

Year:    2018

Author:    Wai Sun Don, Peng Li, Kwun Ying Wong, Zhen Gao

Advances in Applied Mathematics and Mechanics, Vol. 10 (2018), Iss. 6 : pp. 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 two-dimensional 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.OA-2017-0292

Advances in Applied Mathematics and Mechanics, Vol. 10 (2018), Iss. 6 : pp. 1418–1439

Published online:    2018-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    22

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

Author Details

Wai Sun Don

Peng Li

Kwun Ying Wong

Zhen Gao

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