A parameter‐free ε‐adaptive algorithm for improving weighted compact nonlinear schemes

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. We propose a parameter‐free algorithm to calculate ε adaptively based on local flow characteristics for improving WCNS. We discuss the relation between critical points and discontinuities and illustrate that, if the large gradient areas caused by high‐order critical points are not resolved finely enough, numerical oscillations arise. Numerical tests 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.