A Hierarchical Multi‐Resolution <scp>WENO</scp> Scheme for Hyperbolic Conservation Laws

ABSTRACT To achieve the high‐precision characteristics within the smooth regions while maintaining stable, non‐oscillatory, and sharp discontinuity transitions, the weighted essentially non‐oscillatory (WENO) scheme and its variants have been developed. However, when there are multiple discontinuities close to each other, the numerical accuracy and robustness of the traditional schemes are probably affected. To this end, a hierarchical multi‐resolution WENO (MR‐WENO) scheme is proposed in the present study to suppress the overshoot and improve the accuracy of the original MR‐WENO scheme in the vicinity of discontinuities. It could achieve an adaptive selection of the substencils and optimal accuracy due to the hierarchical strategy and the new smoothness indicator. The performances of the hierarchical MR‐WENO scheme have been tested in 1D and 2D cases. The accuracy has been validated and the influences of weights of both large stencils and small substencils have been comprehensively discussed. The linear weights are adjusted aiming at improving the resolution of discontinuities and suppressing the unexpected weight oscillations. As a result, discontinuities like shock waves and contact discontinuities could be accurately resolved, while overshoot phenomena in the MR‐WENO scheme are effectively suppressed in both 1D and 2D cases. Especially, the numerical error of the Lax problem has been reduced by one or two orders of magnitude in the vicinity of discontinuities. With an implementation of the KXRCF indicator, the computational cost has been controlled while maintaining the present superiority of shock capturing capacity.