在這篇文章中，基於Operator-Splitting方法，我們提供一個數值方法來算這個方程的Riemann問題的近似解。這個近似解是由entropy solution和擾動項所組成，entropy solution是解齊次守恆律的Riemann問題所解出來的，擾動項是解一個利用Operator-Splitting方法和平均線性系統中的不連續係數所得到的近似的常微分方程問題。;In this thesis, we consider the compressible Euler-Poisson equations in 2-dimensional space. The equations are in the form of hyperbolic system of balance laws coupled with Poisson equation, which is a mixed-type system of partial differential equations. The mixed-type system describes the conservation of mass, momentum of fluid under the effect of gravitational force, which is one of the most important PDE models in physics, astrophysics and Cosmology. The global existence of solutions to the initial-boundary value problem of the compressible Euler-Poisson equations in 2-dimensional space has been unsolved due to the lack of regularity of solutions caused by the appearance of shock waves. In addition, there is no efficient numerical method of constructing the approximate solutions for the system. In this article, we provide a numerical method for the approximate solution of Riemann problem based on the framework of operator-splitting method. The approximate solution consists of the entropy solution of the Riemann problem of associated homogeneous conservation laws and the perturbation term solving a linearized hyperbolic system with discontinuous coefficients. The perturbation term is obtained by solving an approximate ODEs problem modified by the operator-splitting method and averaging process to the discontinuous coefficients in the linearized hyperbolic system.
