Weights Design For Maximal Order WENO Schemes

Abstract

Weighted essentially non-oscillatory (WENO) finite difference schemes, developed by Liu et al. (Comput Phys 115(1):200---212, 1994) and improved by Jiang and Shu (Comput Phys 126(1):202---228, 1996), are one of the most popular methods to approximate the solutions of hyperbolic equations. But these schemes fail to provide maximal order accuracy near smooth extrema, where the first derivative of the solution becomes zero. Some authors have addressed this problem with different weight designs. In this paper we focus on the weights proposed by Yamaleev and Carpenter (J Comput Phys 228:4248---4272, 2009). They propose new weights to provide faster weight convergence than those presented in Borges et al. (J Comput Phys 227:3191---3211, 2008) and deduce some constraints on the weights parameters to guarantee that the WENO scheme has maximal order for sufficiently smooth solutions with an arbitrary number of vanishing derivatives. We analyze the scheme with the weights proposed in Yamaleev and Carpenter (J Comput Phys 228:4248---4272, 2009) and prove that near discontinuities it achieves worse orders than classical WENO schemes. In order to solve these accuracy problems, we define new weights, based on those proposed in Yamaleev and Carpenter (J Comput Phys 228:4248---4272, 2009), and get some constraints on the weights parameters to guarantee maximal order accuracy for the resulting schemes.