Embedded WENO: a design strategy to improve existing WENO schemes
This work provides a general design strategy to enhance existing WENO schemes for computational fluid dynamics, offering incremental improvements in accuracy and spectral properties.
Embedded WENO methods improve existing WENO schemes by using all adjacent smooth substencils, achieving better spectral properties or higher order of convergence for smooth solutions, with improvements also for discontinuous solutions, all without added computational cost.
Embedded WENO methods utilize all adjacent smooth substencils to construct a desirable interpolation. Conventional WENO schemes under-use this possibility close to large gradients or discontinuities. We develop a general approach for constructing embedded versions of existing WENO schemes. Embedded methods based on the WENO schemes of Jiang and Shu (J. Comput. Phys. 126 (1996)) and on the WENO-Z scheme of Borges et al. (J. Comput. Phys. 227 (2008)) are explicitly constructed. Several possible choices are presented that result in either better spectral properties or a higher order of convergence for sufficiently smooth solutions. However, these improvements carry over to discontinuous solutions. The embedded methods are demonstrated to be indeed improvements over their standard counterparts by several numerical examples. All the embedded methods presented have no added computational effort compared to their standard counterparts.