The Research Of Entropy Stable Schemes For Shallow Water Equations

The numerical methods for shallow water equations are one of the important research topics in computational fluid dynamics.The shallow water equations are a kind of nonlinear hyperbolic conservation laws,and its numerical methods do not guarantee that the solutions are physical.To solve this problem,more and more numerical methods that satisfy the entropy stable conditions have been proposed in recent years,such as entropy stable schemes that were discussed in this thesis.These kind of schemes avoided non-physical phenomenon,such as “dispersion effect” and “expansion shock”,which has a good application prospect in solving hyperbolic conservation laws.The specific work of this paper includes:(1)A high resolution entropy stable scheme which satisfied the Theorem of the Second Law of Thermodynamics is first developed on a rectangular grid of shallow water equations with source terms,and a higher order entropy conservative scheme and a first order entropy stable scheme are combined with a flux limiter function.The new entropy stable scheme have higher accuracy in the regions of the smooth solutions and capture shocks accurately while avoiding non-physical phenomena in the regions of the discontinuous solutions.Finally,the new scheme successfully calculated the classical one-dimensional and two-dimensional problems,and we verified the accuracy of the new scheme.(2)An entropy stable scheme is constructed directly on the triangular grids by using the finite volume methods.Firstly,the semi-discrete scheme of two-dimensional shallow water equations is obtained by using the vertex-centered discrete methods.And the numerical flux of entropy conservative scheme on unstructured grids is constructed.Secondly,the entropy stable scheme is obtained by adding the Roe's dissipative operator on the entropy conservation flux and the theoretical results have been proven.Finally,the triangle grids generated by the Delaunay function are sorted,and the virtual boundaries needed for the computation are added.And the dam break problems of two-dimensional shallow water equations are calculated on the generated grids.The final results show that the entropy stable scheme can successfully simulate the flow of shallow water equations.