Embedded WENO: A design strategy to improve existing WENO schemes

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 [1] and on the WENO-Z scheme of Borges et al.[2] 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.

A high performance fifth‐order multistep WENO scheme

“…Equation (43) 20 and in this paper show that the E-WENO and IM-WENO schemes cannot effectively reduce the dissipation of WENO-Z near the discontinuities. This paper will further develop multistep WENO scheme, which can obtain high-order accuracy in smooth regions and can also decrease the dissipation near the discontinuities.…”

“…However, limited by the condition of ( k ∕ j , k ≠ j), the simple linear combination of different , such as the formula of Ma et al 19 and van Lith et al, 20 cannot achieve the expected effects. However, limited by the condition of ( k ∕ j , k ≠ j), the simple linear combination of different , such as the formula of Ma et al 19 and van Lith et al, 20 cannot achieve the expected effects.…”

“…As indicated above, theoretically, the idea of multistep weighting is helpful for improving the accuracy at transition point. However, limited by the condition of ( k ∕ j , k ≠ j), the simple linear combination of different , such as the formula of Ma et al 19 and van Lith et al, 20 cannot achieve the expected effects. Hence, we propose a new method by using a function h( l , 1 )(l = 0, 2) to replace l + 1 in Equation (28), and the function should satisfy…”

“…Recently, van Lith et al 20 proposed a new method to improve the WENO-Z scheme. Their formula for calculating the weights is…”

“…Compared with the multistep WENO scheme, although the improved scheme can greatly improve the computational efficiency, the numerical results show that it cannot effectively reduce the dissipation near the discontinuities and its accuracy at critical point is of fourth order. Recently, van Lith et al 20 developed a class of embedded WENO scheme. The scheme is simple and provides a choice to improve the accuracy of the WENO-Z scheme at a transition point.…”