A Finite Volume Scheme For Shallow Water Wave Equations Based On Static Water Reconstruction

In this article,we develop new high-order well-balanced finite volume weighted essentially non-oscillatory(WENO)schemes for the shallow water equations.Well-balanced schemes are characterized by maintaining the steady state solutions exactly at the discrete level.The well-balancing property is of paramount importance in the practical applications where many studied phenomena are frequently interpreted as small perturbations to equilibrium states.To achieve the well-balancing property,we propose to construct the numerical fluxes by means of a suitable conservative variables decomposition as well as the hydrostatic reconstruction idea.This decomposition strategy also helps us to achieve a novel simple approximation of the geometrical source term.Both rigorous theoretical analysis and extensive numerical examples all verify that the current schemes maintain the well-balancing property exactly.Furthermore,numerical results strongly suggest that the proposed schemes enjoy the ability to accurately capture small perturbations to the steady state even on relatively coarse mesh and keep the genuine high-order accuracy for smooth solutions at the same time.