Two Classes Of Balanced Methods For Solving Two-layer Shallow Water Equations

Since the two-layer shallow water equation can more accurately simulate the laws of various flows in reality than the single-layer shallow water equation and the multi-dimensional shallow water equation,and it has a better effect in dealing with many practical problems such as geophysical flow,environment and water conservancy engineering.Therefore,it has been widely studied by scholars in recent years.However,because the two-layer shallow water equation is a partial differential equation and its own coupling terms and hyperbolic constraints,it is difficult to directly calculate the exact solution,so the study of its numerical solution becomes particularly important.Therefore the finite volume method WENO scheme and the finite element method RKDG scheme which have wellbalanced property are constructed to solve the two-layer shallow water equations respectively.In this article,a two-step splitting scheme is first selected to deal with the coupling terms in the two-layer shallow water equations,then for the equations obtained after decoupling,based on the idea of the finite volume method,a numerical method for solving the two-layer shallow water equation is constructed using the WENO reconstruction principle,and the spatial accuracy and wellbalanced properties of the method are analyzed and proved.Finally,the numerical accuracy of the method is verified by a numerical example.The results of the numerical example also show that the method meets the actual flow state of the equation when solving the two-layer shallow water equation,which shows the effectiveness of the method.Then for the equations obtained after decoupling,based on the idea of discontinuous Galerkin scheme in the finite element method,this article constructs a RKDG numerical method for solving two-layer shallow water equations.Firstly,the spatial accuracy of the discontinuous Galerkin scheme is proved using the form of Taylor expansion,and then analyzes the well-balanced properties and numerical accuracy of the method.Finally the correctness of the theory is verified by a numerical example.The result of the example also shows that the numerical solution is in accordance with the motion state of the two-layer shallow water equation,indicating that the method is effective.