A new fifth-order finite difference well-balanced multi-resolution WENO scheme for solving shallow water equations. (1st September 2020)

• Record Type: Journal Article
• Authors: Wang, Zhenming; Zhu, Jun; Zhao, Ning
• Is Part Of: Computers & mathematics with applications. Volume 80:issue 5(2020)
• Page Start: 1387
• Page End: 1404
• Publication Date: 2020-09-01
• Subjects: Shallow water equations -- Multi-resolution WENO scheme -- Well-balanced procedure -- Exact C-property -- Convergence property; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2020.07.003
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: In this paper, a new fifth-order finite difference well-balanced multi-resolution weighted essentially non-oscillatory (WENO) scheme is designed to solve for one-dimensional and two-dimensional shallow water equations with or without source terms on structured meshes. We only use the information defined on a hierarchy of nested central spatial stencils without introducing any equivalent multi-resolution representation once again. For the shallow flow problems with smooth or discontinuous bed, we combine with the well-balanced procedure developed by Xing and Shu (2005) for balancing the flux gradients and the source terms, and then the new fifth-order well-balanced multi-resolution WENO scheme (Zhu and Shu, 2018) could satisfy the exact C-property for still stationary solutions, maintain the fifth-order accuracy in smooth regions, and keep essentially non-oscillatory property in non-smooth regions. Compared with the classical well-balanced WENO schemes (Xing and Shu, 2005), the new features of this finite difference well-balanced multi-resolution WENO scheme is its simplicity and hierarchical structure in obtaining higher-order accuracy and its linear weights could be arbitrarily chosen with one requirement that their summation is one. It is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing finite difference well-balanced WENO scheme for solving shallow water equations. This new well-balanced WENO scheme is simple to construct and can be easily implemented to arbitrary high-order accuracy and in higher dimensions. Some benchmark numerical examples are performed to illustrate the good performances of this new well-balanced WENO scheme which could get high-order accuracy, keep exact C-property, sustain good convergence property, and obtain sharp shock transitions in the whole computational field.