Convergence of the Lax–Friedrichs scheme for Euler-Possion
- 用Lax Friedrichs格式构造其Euler-Possion eq 的近似解。利用补偿紧性框架得到\(\gamma=1\)时的收敛性和一致性。得到了\(L_\infty\)的全局熵解。此处处理的是包含无界速度的初始条件,这与等熵情况不同。
对Possion equation 直接使用Green 函数法解出来。Euler-Possion 可以得到
\[\begin{align}
\rho_t+m_x &=0 \\
m_t+(\frac{m^2}{\rho} +\rho )_x&=\rho(\int G_x(\rho-D) d\mu)-\frac{m}{\tau}
\end{align}
\]
初边值条件查看原文。
- Theorem 1.2. Suppose that the initial data \((\rho_0,m_0)\) and the given function \(D(x)\) and \(\tau\) satisfy the following conditions:
\[0\leq \rho_0 \leq C_0,|m_0| \leq \rho_0(C_0+|log \rho_0|),|D(x)| \leq C_0, 0<\tau_0 \leq \tau
\]
- Then the initial-boundary value problem for Euler-Possion has a global weak entropy solution \(\rho,m\) and the following inequalities:
\[0<\rho<M,|m| \leq \rho(M+\log \rho)
\]
as \(0<t<T\)
-
Lemma1 齐次方程Riemann 问题全局解。
-
Lemma2 \(\Lambda={(\rho,m): w\leq w_0, z\geq z_0}\) is an invariant region. which means if. the Riemann data in \(\Lambda\) then the solution in \(\Lambda\) as well.
-
Entropy flux pair The entropy-flux pair \((\eta,q)\) for \(\gamma=1\) are
\[\eta=\rho^{1/(1-\xi^2)}e^{\xi/(1-\xi^2)} m/ \rho , q=(m/\rho+\xi)\rho^{1/(1-\xi^2)}e^{\xi/(1-\xi^2)} m/\rho
\]
- Compact framework
if (1) \(0 \leq \rho^\epsilon \leq C ,|m^\epsilon| \leq (C+|ln \rho^\epsilon|)\)
(2) $$\eta_t(\rho^{\epsilon} ,m^\epsilon) +q_x(\rho^{\epsilon} ,m^\epsilon) $$ is compact in \(H^{-1}_{loc}\)
Then : exist subsequence \((\rho^{\epsilon},m^{\epsilon} ) \to (\rho,m)\) in \(L^p_{loc}\)
参考文献
Li, Tian-Hong. “Convergence of the Lax–Friedrichs scheme for isothermal gas dynamics with semiconductor devices.” Zeitschrift für angewandte Mathematik und Physik ZAMP 57 (2005): 12-32.