Improved Symmetry Property of High Order Weighted Essentially Non-Oscillatory Finite Difference Schemes for Hyperbolic Conservation Laws (Advances in Applied Mathematics and Mechanics, 2021)
This study aims to investigate the rapid loss of numerical symmetry forproblems with symmetrical initial conditions and boundary conditions when solvedby the seventh and higher order nonlinear characteristic-wise weighted essentiallynon-oscillatory (WENO) finite difference schemes. Using the one-dimensional doublerarefaction wave problem and the Sedov blast-wave problems, and the two-dimensionalRayleigh-Taylor instability (RTI) problem as examples, we illustrate numericallythat the sensitive interaction of the round-off error due to the numerical unstableexplicit form of the local lower order smoothness indicators in the nonlinearweights definition, which are often given and used in the literature, and the nonlinearityof the WENO scheme are responsible for the rapid growth of asymmetry of anotherwise symmetric problem. An equivalent but compact and numerical stable compactform of the local lower order smoothness indicators is suggested for delaying theonset of and reducing the magnitude of the symmetry error. The benefits of using thecompact form of the local lower order smoothness indicators should also be applicableto non-symmetrical strongly non-linear problems in terms of improved numericalstability, reduced rounding errors and increased computational efficiency.