A well-balanced conservative high-order alternative finite difference WENO (A-WENO) method for the shallow water equations

In this paper, we develop a well-balanced, conservative, high-order finite difference weighted essentially non-oscillatory (WENO) method for the shallow water equations. Our approach exactly preserves the moving-water equilibria of the shallow water equations with non-flat bottom topography. The proposed method consists of two key components. First, we reformulate the source term into a flux-gradient form and discretize it using the same numerical flux as that of the true flux gradient to achieve the well-balanced property. Second, we interpolate the equilibrium variables, which remain constant at steady state, to construct the numerical flux. To achieve high-order accuracy and avoid truncation errors when obtaining equilibrium variables, we build our scheme within the alternative finite difference WENO (A-WENO) framework, which operates on point values rather than cell averages. Special attention is given to ensure that the conservation property is not compromised when designing well-balanced discretizations for the source term. We carefully analyze potential causes of non-conservative schemes in the discretization and explain why the discretized source term in our method is both conservative and simple. Extensive numerical tests are presented to validate the performance of the proposed method.