TY - JOUR
T1 - Improved Symmetry Property of High Order Weighted Essentially Non-Oscillatory Finite Difference Schemes for Hyperbolic Conservation Laws
AU - Don , Wai Sun
AU - Li , Peng
AU - Wong , Kwun Ying
AU - Gao , Zhen
JO - Advances in Applied Mathematics and Mechanics
VL - 6
SP - 1418
EP - 1439
PY - 2018
DA - 2018/09
SN - 10
DO - http://doi.org/10.4208/aamm.OA-2017-0292
UR - https://global-sci.org/intro/article_detail/aamm/12717.html
KW - Weighted essentially non-oscillatory, symmetry, smoothness indicator, hyperbolic
conservation laws.
AB - <p style="text-align: justify;">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.</p>