High Order Accurate WENO Numerical Simulation Of Shallow Water Equation Based On Finite Difference Method

China is one of the countries in the world with the most complex water situations,the greatest difficulty in river management,and the heaviest water management tasks.Issues such as dam breaches,tides,and water environments all fall under the category of shallow water flow problems.Researching the shallow water equations not only promotes the long-term development of intelligent water conservancy and modernization of water infrastructure but also holds significant importance for current water engineering construction..The essence of the shallow water equations lies in the hyperbolic conservation laws.The characteristic of their solutions is that regardless of whether the initial conditions are smooth or not,the solutions of the equations will experience discontinuities as time progresses.Low-accuracy numerical schemes are incapable of accurately simulating the behavior of solutions to hyperbolic conservation laws.Therefore,it is necessary to establish numerical schemes that can suppress numerical oscillations,possess high accuracy,high resolution,and effectively capture discontinuities.In this paper,based on the Weighted Essentially Non-Oscillatory(WENO)finite difference scheme,we study the smoothness factors in the WENO-Z scheme.Firstly,by decomposing and Taylor expanding the global smoothness factors and recombining them with the smoothness factors on small templates,a new third-order WENO scheme(WENO-New)for solving the shallow water equations is established by introducing a parameter that regulates dissipation.Secondly,the WENO-JS scheme,WENO-Z scheme,and WENO-New scheme are used to compute classic numerical examples,and the results are compared.The accuracy of the schemes is verified using smooth initial conditions for linear advection equations,while the resolution of the schemes is validated using discontinuous initial conditions for linear advection equations and inviscid Burgers’ equation.The numerical experimental results demonstrate that,under the same number of grids,the calculation errors of the new scheme are smaller than those of the WENO-Z and WENO-New schemes,and the new scheme maintains third-order accuracy at the extrema of functions and exhibits strong modeling capabilities for discontinuity problems.Finally,numerical computations are performed on shallow water equations with and without source terms,and the results indicate that the scheme is computationally stable and possesses excellent performance in capturing shock waves,thereby providing significant reference value for numerical simulations.