Finite difference alternative unequal-sized weighted essentially non-oscillatory schemes for hyperbolic conservation laws

Abstract

In this paper, two unequal-sized weighted essentially non-oscillatory (US-WENO) schemes are proposed for solving hyperbolic conservation laws. First, an alternative US-WENO (AUS-WENO) scheme based directly on the values of conserved variables at the grid points is designed. This scheme can inherit all the advantages of the original US-WENO scheme [J. Zhu and J. Qiu, “A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws,” J. Comput. Phys. 318, 110–121 (2016).], such as the arbitrariness of the linear weights. Moreover, this presented AUS-WENO scheme enables any monotone fluxes applicable to this framework, whereas the original US-WENO scheme is only suitable for the more dissipative smooth flux splitting. Therefore, the method in this paper has a smaller L1 and L∞ numerical errors than the original scheme under the same conditions. Second, in order to further improve the computational efficiency of the above AUS-WENO scheme, a hybrid AUS-WENO scheme is proposed by combining a hybrid strategy. This strategy identifies the discontinuous regions directly based on the extreme points of the reconstruction polynomial corresponding to the five-point stencil, which brings the important advantage that it does not depend on the specific problem and does not contain any artificial adjustable parameters. Finally, the performance of the above two AUS-WENO schemes in terms of low dissipation, shock capture capability, discontinuity detection capability, and computational efficiency is verified by some benchmark one- and two-dimensional numerical examples.