Generalized Riemann Solutions to Compressible Euler-Poisson Equations in Two-dimensional Space

以作者查詢圖書館館藏

、以作者查詢臺灣博碩士

、以作者查詢全國書目

、勘誤回報

、線上人數：26

、訪客IP：18.116.51.45

姓名

楊鈞硯(Jun-Yen Yang) 查詢紙本館藏

畢業系所

數學系

論文名稱

(Generalized Riemann Solutions to Compressible Euler-Poisson Equations in Two-dimensional Space)

相關論文

★ 氣流的非黏性駐波通過不連續管子之探究	★ An Iteration Method for the Riemann Problem of Some Degenerate Hyperbolic Balance Laws
★ 影像模糊方法在蝴蝶辨識神經網路中之應用	★ 單一非線性平衡律黎曼問題廣義解的存在性
★ 非線性二階常微方程組兩點邊界值問題之解的存在性與唯一性	★ 對接近音速流量可壓縮尤拉方程式的柯西問題去架構區間逼近解
★ 一些退化擬線性波動方程的解的性質.	★ 擬線性波方程中片段線性初始值問題的整體Lipchitz連續解的
★ 水文地質學的平衡模型之擴散對流反應方程	★ 非線性守恆律的擾動Riemann 問題的古典解
★ BBM與KdV方程初始邊界問題解的週期性	★ 共振守恆律的擾動黎曼問題的古典解
★ 可壓縮流中微黏性尤拉方程激波解的行為	★ 非齊次雙曲守恆律系統初始邊界值問題之整域弱解的存在性
★ 有關非線性平衡定律之柯西問題的廣域弱解	★ 單一雙曲守恆律的柯西問題熵解整體存在性的一些引理

檔案

[Endnote RIS 格式]

[Bibtex 格式]

[相關文章]

[文章引用]

[完整記錄]

[館藏目錄]

[檢視]

[下載]

本電子論文使用權限為同意立即開放。
已達開放權限電子全文僅授權使用者為學術研究之目的，進行個人非營利性質之檢索、閱讀、列印。
請遵守中華民國著作權法之相關規定，切勿任意重製、散佈、改作、轉貼、播送，以免觸法。

摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ Euler-Poisson方程

關鍵字(英)

★ Riemann Solver
★ Euler-Poisson equation
★ Euler-Poisson equation in 2D
★ Splitting method

論文目次

Contents
中文摘要i
Abstract ii
Contents iii
1 Introduction 1
2 Operator-splitting Method for Euler-Poisson Equations 5
3 Construction of Approximate Solutions to (2:6) and (2:7) 10
3.1 Construct of Approximate Solution of (2:6) . . . . . . . . . . . . . . . . . . . 10
3.2 Construct of Approximate Solution of (2:7) . . . . . . . . . . . . . . . . . . . 16
3.3 Construction of the perturbation term U
. . . . . . . . . . . . . . . . . . . . . 19
4 Steady States 20

參考文獻

[1] 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.
[2] S.W. Chou, J.M. Hong, Y.C. Su, Global entropy solutions of the general nonlinear hyperbolic
balance laws with time-evolution flux and source, Mathods Appl. Anal., 19 (2012),
pp. 43–76.
[3] S.W. Chou, J.M. Hong, Y.C. Su, The initial-boundary value problem of hyperbolic integrodifferential
systems of nonlinear balance laws, Nonlinear Anal. 75 (2012), pp. 5933-5960.
[4] C.M. Dafermos, L. Hsiao, Hyperbolic systems of balance laws with inhomogeneity and
dissipation, Indiana Univ. Math. J. 31 (1982), pp. 471-491.
[5] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Commun.
Pure Appl. Math. 18 (1965), pp. 697-715.
[6] J. B. G??????, Initial boundary value problems for hyperbolic systems of conservation
laws, Thesis (Ph. D.)–Stanford University., (1983).
[7] J. Groah, J. Smoller, B. Temple, Shock Wave Interactions in General Relativity, Monographs
in Mathematics, Springer, Berlin, New York, 2007.
[8] 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.
Diff. Equ. 222 (2006), pp. 515-549.
[9] J.M. Hong, P.G. LeFloch, A version of Glimm method based on generalized Riemann
problems, J. Portugal Math. 64, (2007) pp. 199-236.
[10] 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.
[11] E. Isaacson, B. Temple, Nonlinear resonance in systems of conservation laws, SIAM J.
Appl. Anal. 52 (1992), pp. 1260-1278.
[12] P.D. Lax, Hyperbolic system of conservation laws II, Commun. Pure Appl. Math. 10
(1957), pp. 537-566.
[13] 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.
[14] T.-P. Liu, Quasilinear hyperbolic systems, Commun. Math. Phys. 68 (1979), pp. 141-172.
[15] 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.
[16] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd ed., Springer-Verlag,
Berlin, New York, 1994.
[17] 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.
[18] John M. Hong, Reyna Marsya Quita, Approximation of Generalized Riemann Solutions
to Compressible Euler-Poisson Equations of Isothermal Flows in Spherically Symmetric
Space-times (2016)

指導教授

洪盟凱(John M. Hong)

審核日期

2017-7-3

推文