在這個問題中，我們考慮一個在二維時空下可壓縮的Euler-Poisson方程組。這個方程組是由守恆律和Poisson方程式組合在一起的hyperbolic系統，它是一個混合型的偏微分方程組。這個方程組是在描述流體的質量和動量在重力的影響下的守恆性，不管是在物理、天體物理、還是宇宙學中，它都是非常重要的偏微分方程模型。此方程的初值邊界問題的解，由於震波的發生而導致缺乏解的規律性，使得找不到解的全域存在性。此外，也沒有一個好的數值方法來建構此方程的近似解。

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.

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