Research On Entropy Stabilization Algorithm Of Rotating Flux For Shallow Water Equation

Hyperbolic conservation laws equation is one of the most important ones in computational mathematics field(including inviscid Burger's equation,shallow water equation,Euler equation and magnetohydrodynamic equation).As we all know,the numerical results of this kind of equations are not only related to the scheme in space,but also to the grid distribution(such as structural grid and unstructured grid).Therefore,it is the core of how to choose the numerical scheme and how to segment the grid.The entropy stable scheme satisfies the second law of thermodynamics.At present,the scheme adds the artificial viscosity term on the basis of entropy conservation scheme which has small viscosity in the smooth region,which does not affect the high accuracy of the scheme in the smooth region,and also ensures the numerical stability of the scheme in the discontinuous region.The advantage of the moving mesh method is the one which keeps the number of grid nodes no change,make the grid automatically move to the area where the nature of the solution changes greatly(such as the discontinuous area),and move away the area where the nature of the solution changes little.This method can improve the resolution of the numerical resultsFor the two-dimensional shallow water equation,the rotation invariance,which makes the two-dimensional problem be similar to one-dimensional for the rotation flux and ensures that all sides of the grid element move in the same direction at the same time with the grid movement,to reduce the dependence on the direction of the grid,to avoid the grid winding in the use of the two-dimensional moving grid,so that the calculation can not be carried out.To solve the shallow water equation by coupling the moving grid method and The rotated flux entropy stable scheme based on the moving mesh method is studied to ensure the high accuracy of the entropy stable scheme,and to improve the resolution of the discontinuity.The main research contents are as follows(1)Because the moving mesh method has the characteristics of keeping the number of nodes unchanged and of being able to move adaptively to the discontinuous area when dealing with discontinuous problems,this paper couples the one-dimensional moving grid method with the one-dimensional entropy stable scheme to solve the one-dimensional shallow water equation,and compares the numerical results with the reference solution and the fixed grid entropy stable scheme.The results show that the new algorithm improves the efficiency of the algorithm.The resolution at the discontinuity is improved without nonphysical phenomenon(2)The two-dimensional moving grid method is used to solve the two-dimensional shallow water equation.The numerical simulation results show oscillation and the calculation cannot be carried out.Rotation invariance has the feature of quasi one-dimensional,which can reduce the dependence on grid direction.Based on the fixed grid,the rotated flux is derived by using rotation invariance of flux function's,which is combined with entropy stable scheme to solve the two-dimensional shallow water equation.Compared with the fixed grid entropy stable scheme,the numerical results show that the fixed grid entropy stable scheme can simulate the small changes for the continuous case,and has the characteristics of high resolution and good symmetry(3)The moving mesh method can deal with the discontinuity problem well and improve the resolution of the discontinuity.When the moving grid method in(1)is extended to the two-dimensional shallow water equation,and some of the examples occur oscillation,the calculation can not be carried out.We try to combine the rotated flux in(2)with the two-dimensional moving grid method to get a new algorithm:the moving grid rotated flux entropy stable scheme method,for solve the two-dimensional shallow water equation.The numerical examples,show that the numerical results using the new algorithm haave good symmetry,no oscillation and high resolution.