Premium
A parameter‐free ε ‐adaptive algorithm for improving weighted compact nonlinear schemes
Author(s) -
Zheng Shichao,
Deng Xiaogang,
Wang Dongfang,
Xie Chunhui
Publication year - 2019
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.4719
Subject(s) - classification of discontinuities , mathematics , nonlinear system , smoothness , dissipation , spurious relationship , algorithm , numerical analysis , convergence (economics) , order of accuracy , computer simulation , interpolation (computer graphics) , discontinuity (linguistics) , mathematical analysis , numerical stability , computer science , physics , statistics , quantum mechanics , thermodynamics , economic growth , animation , computer graphics (images) , economics
Summary In this paper, we propose a parameter‐free algorithm to calculate ε , a parameter of small quantity initially introduced into the nonlinear weights of weighted essentially nonoscillatory (WENO) scheme to avoid denominator becoming zero. The new algorithm, based on local smoothness indicators of fifth‐order weighted compact nonlinear scheme (WCNS), is designed in a manner to adaptively increase ε in smooth areas to reduce numerical dissipation and obtain high‐order accuracy, and decrease ε in discontinuous areas to increase numerical dissipation and suppress spurious numerical oscillations. We discuss the relation between critical points and discontinuities and illustrate that, when large gradient areas caused by high‐order critical points are not well resolved with sufficiently small grid spacing, numerical oscillations arise. The new algorithm treats high‐order critical points as discontinuities to suppress numerical oscillations. Canonical numerical tests are carried out, and computational results indicate that the new adaptive algorithm can help improve resolution of small scale flow structures, suppress numerical oscillations near discontinuities, and lessen susceptibility to flux functions and interpolation variables for fifth‐order WCNS. The new adaptive algorithm can be conveniently generalized to WENO/WCNS with different orders.