Embedded WENO: a design strategy to improve existing WENO schemes

TL;DR

This paper introduces embedded WENO schemes that leverage all adjacent smooth substencils to enhance accuracy and spectral properties near discontinuities without increasing computational cost.

Contribution

The authors develop a general framework for creating embedded WENO schemes based on existing methods, improving accuracy and spectral properties near discontinuities.

Findings

Embedded WENO schemes outperform standard schemes in numerical tests.

No additional computational effort is required for embedded schemes.

Improvements are observed for both smooth and discontinuous solutions.

Abstract

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.