A new fifth-order finite difference well-balanced multi-resolution WENO scheme for solving shallow water equations

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.