# 臺灣博碩士論文加值系統

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 對於在跨音速流下的一維可壓縮無源項的尤拉方程，因為沒有 $a(x, t)$ 的影響，其解可以用 Lax 的方法構造。如果 $\tilde{U}$ 是黎曼解，我們可得到尤拉系統對 $\tilde{U}$ 展開的線性方程：\begin{align*}&\bar{U}_t+(DF(\tilde{U})\bar{U})_x\\&=-\frac{a_x}{a}(G_1(\tilde{U})+DG_1(\tilde{U})\bar{U})-\frac{a_t}{a}(G_2(\tilde{U})+DG_2(\tilde{U})\bar{U})\end{align*}當 $\tilde{U}$ 是一個衝擊波時，此方程的係數是不連續的，所以為了簡化問題，我們對方程的時間 $t$ 平均係數。因此，一個逼近解 $\overline{U}$ 可以由常微分方程的理論得到。為構建原始尤拉系統初值問題的一個局部逼近解，我們打算利用算子裂解法。此法常用於構造雙曲守恆律的逼近解。因此，一個逼近解是由傳統黎曼解和微擾的疊加所構成：$$U=\tilde{U}+\bar{U}$$
 For one dimensional full compressible Euler equations with no source in transonic flow, the solution is constructed by Lax's method because there is no influence of $a(x,t)$. If $\tilde{U}$ is the Riemann solution, we obtain a linearized equations of Euler system around $\tilde{U}$:\begin{align*}&\bar{U}_t+(DF(\tilde{U})\bar{U})_x\\&=-\frac{a_x}{a}(G_1(\tilde{U})+DG_1(\tilde{U})\bar{U})-\frac{a_t}{a}(G_2(\tilde{U})+DG_2(\tilde{U})\bar{U})\end{align*}Since the coefficients of this equation are discontinuous when $\tilde{U}$ is a shock wave, to simplify the problem we average the coefficients with respect to time $t$. Hence, an approximate solution $\overline{U}$ can be obtained by the theory of o.d.e.To construct a local approximate solution to the initial value problem of the original Euler system, we utilize the scheme of operator splitting which is used frequently in constructing approximate solutions of the conservation laws. Therefore, an approximate solution is constructed as a superposition of the classical Riemann solution and the perturbation:$$U=\tilde{U}+\overline{U}$$
 Contents1 Introduction 52 The Riemann problem of the homogeneous Euler system 83 Operator splitting scheme to the Euler system 123.1 The solution to the linear first order system (3.16) 213.2 Another approximate solution to the linear first order system (3.16) 283.3 The comparison of the solutions to (3.16) and (3.53) 29A The Riemann solver of the Euler system in a uniform duct 32B The proof of Theorem 3.5 41
 [1] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,Comm. Pure Appl. Math. 18 (1965), pp. 697-715.[2] Chia-Chun Lai, The construction of local approximate solutions to the cauchyproblem of compressible Euler equations in transonicow, Master Thesis, Na-tional Central University, (2007).[3] P.D. Lax, Hyperbolic system of conservation laws II, Comm. Pure Appl. Math.10 (1957), pp. 537-566.[4] J.M. Hong, P.G. LeFloch, A version of Glimm method based on generalizedRiemann problems, J. Portugal Math. 64 (2007) pp. 199-236.[5] T.-P. Liu, The Riemann problem for general systems of conservation laws, J.Di . Eqns 18 (1975) pp. 218-234.[6] J. Smoller, Shock Waves and Reaction-Di usion Equations, 2nd ed., Springer-Verlag, Berlin, New York, 1994.[7] Eleuterio F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics,3rd ed., Springer-Verlag, Berlin Heidelberg, New York, 2009.
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