High-precision Discontinuous Galerkin Finite Element Method For Euler Equations With Isothermal And Steady Solution

In this paper,a high-precision well-balanced Runge-Kutta discontinuous Galerkin(abbreviated as RKDG)method that maintains a steady state solution is studied for the Euler equations that maintain isothermal equilibrium in a gravitational field.First,by analyzing the Euler equation in one-dimensional space under the gravitational field,it can be deduced that the Euler equation has a steady solution with zero velocity.Then,according to the situation in practical application,this paper briefly introduces two equilibrium states,namely isothermal hydrostatic state and isentropic hydrostatic state,and this paper only considers the Euler equation of isothermal hydrostatic state.Then,the basic principle and process of the Discontinuous Galerkin(abbreviated as DG)method are briefly reviewed,and the high-precision RKDG method is designed for the Euler equation that maintains the isothermal equilibrium state under gravitational field.For euler equation with gravity source term,the unknown solution is decomposed into the sum of equilibrium state and remaining state by constructing reasonable auxiliary function,then the numerical fluxes are constructed based on hydrostatics reconstruction idea,and a numerical method with well-balanced property is proposed.Then,with the help of the decomposition algorithm,it is applied to the source term,and the gravity source term is discretized,so that the discretization for the source term and the discretization for the flux gradient can reach a state of mutual balance,and then the well-balanced half can be obtained.The discrete method is to realize the discretization of Euler’s equation in space.The traditional third order Runge-Kutta time discretization method is used to advance the euler equation,and then the RKDG method with high accuracy for the steady state solution can be realized.Then,the RKDG method of one-dimensional Euler equation is extended to the multidimensional space,and it can be verified that the method can keep the well-balanced property accurately in the multidimensional space both theoretically and numerically.Finally,in chapter 4through some numerical examples to verify the superiority of the research method,it can be seen that the method is to keep steady solution to the machine precision,and in a one-dimensional and two-dimensional space numerical example is given respectively,and illustrates flux gradient and the importance of balance between the source term,and in view of the smooth solution to maintain the high accuracy,strong to keep sharp jumps in transition.