A numerical study of the local monotone polynomial edge detection for the hybrid WENO method

The weighted essentially non-oscillatory (WENO) method is a high order numerical method that can handle discontinuous problems efficiently. The hybrid WENO method adopts the WENO reconstruction in the region where the jump discontinuity exists, while other high order reconstruction is adopted in the smooth region. Thus for the hybrid WENO method, the jump location should be identified before the adoption. Various edge detection methods have been developed mostly focusing on finding edges as accurately as possible utilizing as many stencils as possible. However, if the reconstruction in the smooth area is obtained with the fixed finite number of cells, it would suffice to examine whether the considered stencils need to adopt the WENO reconstruction instead of finding edges in a global manner. In this note, we compare the multi-resolution analysis with the local monotone polynomial method for the 5th order hybrid WENO method. The monotone method uses the cell information within the given stencil only to decide whether to adopt the WENO reconstruction or not. In this sense, the local method is optimal. We provide a detailed numerical study and show that the monotone polynomial method is efficient and accurate.