在本文中，我們考慮球對稱空間時間可壓縮歐拉方程泊松。方程，代表品質和重力吸引潛在的物理量動量守恆，可以寫成一個混合型的3x3部分迪系統的差動或平衡法與全球源2X2雙曲系統。我們展示的方程為品質守恆，歐拉 - 泊松下方程可以轉化為平衡定律與本地源純3x3的雙曲系統。歐拉 - 泊松方程黎曼問題，這是在初始邊值問題廣義Glimm方案的構建塊的通用解決方案，提供不嚴的類型相關聯的同質守恆定律弱解和擾動項解決了疊加通過與些線性雙曲系統與這種鬆懈的解決方案。最後，我們提供LAX-Wendroff無限迪方法和辛普森的數值積分為某些初始邊值問題全球資源。提供了幾種類型的初始和邊界資料的數值類比。

In this thesis, we consider the compressible
Euler-Poisson equations in spherically symmetric space-times. The
equations, which represent the conservation of mass and momentum
of physical quantity with attracting gravity potential, can be
written as a mixed-type $3\times 3$ partial differential systems
or an $2\times 2$ hyperbolic systems of balance laws with $global$
source. We show under the equation for the conservation of mass,
Euler-Poisson equations can be transformed into a pure $3\times 3$
hyperbolic system of balance laws with $local$ source. The
generalized solutions to the Riemann problem of Euler-Poisson
equations, which is the building block of generalized Glimm scheme
for the initial-boundary value problem, are provided as the
superposition of Lax's type weak solutions of associated
homogeneous conservation laws and the perturbation terms solved by
some linearized hyperbolic system with coefficients related to
such Lax's solution. Finally, we provide Lax-Wendroff finite
difference method and Simpson's numerical integration to the
global sources for some initial-boundary value problems. Numerical
simulations are provided for several types of initial and boundary
data.

[1] G.-Q. Chen, M. Slemrod, D. Wang, Vanishing viscosity method for transonic flow, Arch. Rational Mech. Anal. 189 (2008), pp. 159-188.
[2] S.W. Chou, J.M. Hong, Y.C. Su, An extension of Glimm's method to the gas dynamical model of transonic flows, Nonlinearity 26 (2013), pp. 1581-1597.
[3] S.W. Chou, J.M. Hong, Y.C. Su, Global entropy solutions of the general non-linear hyperbolic balance laws with time-evolution flux and source, MathodsAppl. Anal., 19 (2012), pp. 43
[4] S.W. Chou, J.M. Hong, Y.C. Su, The initial-boundary value problem of hyperbolic integro-dierential systems of nonlinear balance laws, Nonlinear Anal. 75 (2012), pp. 5933-5960.
[5] C.M. Dafermos, L. Hsiao, Hyperbolic systems of balance laws with inhomogeneity and dissipation, Indiana Univ. Math. J. 31 (1982), pp. 471-491.
[6] G. Dal Maso, P. LeFloch, F. Murat, Denition and weak stability of nonconservative products, J. Math. Pure Appl. 74 (1995), pp. 483-548.
[7] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Commun. Pure Appl. Math. 18 (1965), pp. 697-715.
[8] J. B. Goodman, Initial boundary value problems for hyperbolic systems of conservation laws, Thesis (Ph. D.){Stanford University., (1983).
[9] P. Goatin, P.G. LeFloch, The Riemann problem for a class of resonant nonlinear systems of balance laws, Ann. Inst. H. Poincare-Analyse Non-lineaire 21 (2004), pp. 881-902.
[10] J. Groah, J. Smoller, B. Temple, Shock Wave Interactions in General Relativity, Monographs in Mathematics, Springer, Berlin, New York, 2007.
[11] J.M. Hong, An extension of Glimm's method to inhomogeneous strictly hyperbolic systems of conservation laws by \weaker than weak" solutions of the Riemann problem, J. Di. Equ. 222 (2006), pp. 515-549.
[12] J.M. Hong, P.G. LeFloch, A version of Glimm method based on generalized Riemann problems, J. Portugal Math. 64, (2007) pp. 199-236.
[13] J.M. Hong, B. Temple, The generic solution of the Riemann problem in a neighborhood of a point of resonance for systems of nonlinear balance laws, Methods Appl. Anal. 10 (2003), pp. 279-294.
[14] J.M. Hong, B. Temple, A bound on the total variation of the conserved quantities for solutions of a general resonant nonlinear balance law, SIAM J. Appl. Math. 64 (2004), pp. 819-857.
[15] J.M. Hong, Y.-C. Su, Generalized Glimm scheme to the initial-boundary value problem of hyperbolic systems of balance laws, Nonlinear Analysis: Theory, Methods and Applications 72 (2010), pp. 635-650.
[16] E. Isaacson, B. Temple, Nonlinear resonance in systems of conservation laws, SIAM J. Appl. Anal. 52 (1992), pp. 1260-1278.
[17] E. Isaacson, B. Temple, Convergence of the 2  2 Godunov method for a general resonant nonlinear balance law, SIAM J. Appl. Math. 55, No. 3 (1995), pp. 625-640.
[18] P.D. Lax, Hyperbolic system of conservation laws II, Commun. Pure Appl. Math. 10 (1957), pp. 537-566.
[19] P.G. LeFloch, Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form, Commun. Part. Di. Equ. 13 (1988) pp. 669-727.
[20] P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form, Institute for Math. and its Appl., Minneapolis, Preprint 593, 1989.
[21] P.G. LeFloch, T.-P. Liu, Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum Math. 5 (1993), pp. 261-280.
[22] P.G. LeFloch, P.A. Raviart, Asymptotic expansion for the solution of the generalized Riemann problem, Part 1, Ann. Inst. H. Poincare, Nonlinear Analysis 5 (1988) pp. 179-209.
[23] Randall J. LeVeque, Numerical Methods for Conservation Laws, Basel, Birkhauser Verlag, 1992.
[24] T.-P. Liu, Quasilinear hyperbolic systems, Commun. Math. Phys. 68 (1979), pp. 141-172.
[25] T.-P. Liu, Nonlinear stability and instability of transonic flows through a nozzle, Commun. Math. Phys. 83 (1982), pp. 243-260.
[26] T.-P. Liu, Nonlinear resonance for quasilinear hyperbolic equation, J. Math. Phys. 28 (1987), pp. 2593-2602.
[27] M. Luskin and B. Temple, The existence of global weak solution to the nonlinear waterhammer problem, Commun. Pure Appl. Math. 35 (1982), pp. 697-735.
[28] C.S. Morawetz, On a weak solution for a transonic
ow problem, Commun. Pure Appl. Math. 38 (1985), pp. 797-817.
[29] J. Smoller, On the solution of the Riemann problem with general stepdata for an extended class of hyperbolic system, Mich. Math. J. 16, pp. 201-210.
[30] J. Smoller, Shock Waves and Reaction-Diusion Equations, 2nd ed., Springer-Verlag, Berlin, New York, 1994.
[31] B. Temple, Global solution of the Cauchy problem for a class of 22 nonstrictly hyperbolic conservation laws, Adv. Appl. Math. 3 (1982), pp. 335-375.
[32] N. Tsuge, Existence of global solutions for isentropic gas
flow in a divergent nozzle with friction, J. Math. Anal. Appl. 426 (2015), pp. 971-977.