A New Finite Difference Well-Balanced Mapped Unequal-Sized WENO Scheme for Solving Shallow Water Equations

Abstract

In this paper, we propose a newly designed fifth-order finite difference well-balanced mapped unequal-sized weighted essentially non-oscillatory (WBMUS-WENO) scheme for simulating the shallow water systems on multi-dimensional structured meshes. We design new non-linear weights and a new mapping function, so that the WBMUS-WENO scheme can maintain fifth-order accuracy with a small $ε$ even nearby the extreme points in smooth regions. The truncation errors of the scheme is smaller and it has better convergence in simulating some steady-state problems. Unlike the traditional well-balanced WENO-XS scheme [29], this new WBMUS-WENO scheme uses three unequal-sized stencils, denotes the linear weights to be any positive numbers on condition that their summation is one. By incorporating a quartic polynomial on the whole big stencil into WENO reconstruction, the WBMUS-WENO scheme is simple and efficient. Extensive examples are performed to testify the exact C-property, absolute convergence property, and good representations of this new WBMUS-WENO scheme.