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RSS订阅转载 Test 1D Degenerate Elliptical equation without Hamilton-Jacobi Part
Test 1D Degenerate Elliptical equation without Hamilton-Jacobi Parti 根据上一篇的经验：\(\frac{u_x^2}{1+u_x^2}\) 对初值的选取是很敏感的，建议取消这一项，改成已知部分。 以下测试 \[ \begin{align} u_t &=\frac{u_x}{\sqrt{1+u_x^2}}+S(...
2019-10-04 21:07:00
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转载 Hamilton Jacobi
Hamilton Jacobi 使用的不同的 Flux 和不同的边界条件测试了 \[ u_t=\frac{u_x^2}{1+u_x^2}-\frac{\cos(x)^2}{1+\cos(x)^2},x\in [0,4\pi] \] 最有趣的是测试了初值 \[ u= \left\{ \begin{array}{c} \sin(x),&x\in[0,\pi] \cup [3\p...
2019-10-04 16:12:00
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转载 Inverse Lax Wendroff Procedure
Inverse Lax Wendroff Procedure 为了简单的理解ILW的思想，考虑如下的一维的守恒律： \[ \begin{align} u_t+f(u)_x &=0, x\in(-1,1), t>0\\ u(-1,t) &=g(t), t>0\\ u(x,0)&=u_0(x), x \in[-1,1] \end{align} \] 假...
2019-10-03 22:21:00
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转载 Test code for 1D
Test code for 1D In order to test the WENO5 method and the 5 point center difference method, we test follow simple case \[ \begin{align} u_t&=u_{xx}+\sin(x),x\in [0,2\pi]\\ u_x|_{x=0} &...
2019-10-02 19:14:00
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转载 Jump Test
Jump test We consoder a function \[ f_x=\frac{1}{\varepsilon^2 +(x-x_0)^2} \] Integrate of this function is \[ f=-\frac{1}{\varepsilon}arctan( \frac{x_0-x}{\varepsilon}) \] A good idear ...
2019-09-28 12:07:00
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转载 Jang Equation In Spherical Symmetry Case
Jang Equation In Spherical Symmetry Case Let us assume that the metric is \((\gamma_{rr},\gamma_{\theta\theta},\gamma_{\phi\phi})\), then we calculate the Mean curvature and trace of extrinsic ...
2019-09-26 16:34:00
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转载 Why the Place that Jang'S Equation Blow Up is Apparent Horiozn?
Why the Place that Jang'S Equation Blow Up is Apparent Horiozn? Let's consider a simple example: Let us assume that the three metric is \[ds^2=g_{rr}dr^2+R^2(r)d\Omega\] extrinsic curvature is ...
2019-09-18 15:41:00
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转载 ADM Mass
This total energy is called the ADM mass of slice \(\Sigma_t\) \[ \displaystyle \fbox{$ M_{\rm ADM} = {\dfrac{1}{16\pi}} \lim\limits_{{{S}}_{t}\rightarrow\infty} \displaystyle\oint_{{{S}}_{t}} \...
2019-09-17 22:01:00
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转载 Finding Black Holes 2
Spherically Symmetric Case For the spherically symmetric case, \(f\) is constant. Thus \[ \begin{equation} D_as^a=\frac{1}{\sqrt{\gamma}}\partial_r (\sqrt{\gamma}s^r) \end{equation} \] Becau...
2019-09-17 21:28:00
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转载 背景设置
body { font-family: "仿宋","FangSong", "宋体", "Segoe UI", Tahoma, Arial; font-size: 20px!important; color:blue; } DIV.post DIV.entry { font-family: "仿宋","FangSong", "宋体", "Sego...
2019-09-17 20:00:00
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转载 Finding Black Holes 1
Finding Black Holes 1 The apparent horizon (i.e., the marginally trapped outer surface) is an invaluable tool for finding black holes in numerical relativity: In numerical relativity, the exist...
2019-09-17 17:06:00
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转载 Levenberger-Marquardt for nonlinear elliptical system
考虑如下的方程组，测试Levenberger-Marquardt 方法： \[ \begin{align*} \varphi_{rr}+\frac{2}{r}\varphi_{,r}+\frac{1}{8}(A)^2-\frac{1}{12}\varphi^5 &=f_1\\ A_{,r}-\frac{2}{3}\varphi^6+A &=f_2\\ ...
2019-09-16 10:30:00
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转载 Levenberg-Marquardt method for nonlinear elliptical equation
使用 Levenberg-Marquardt 方法测试 \[u_{xx}+u^2=-sin(x)+sin(x)^2\], 初值选取 \(u(x)=cos(x)sin(x)\) 参考文献：《非线性方程组数值方法》，袁亚湘； 给出初值 \(x_0 \in \mathbb{R}^n; k=1,\eta \in(0,1)\) 若 \(\|J_k^{T} F_k\|=0\), 停; 求解 \(...
2019-09-15 21:49:00
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转载 Finding Peace in 21st-Century Kyoto
When travelers arrive in Kyoto for the first time, they often are confused and disappointed. Expecting a place that exudes timeless elegance and peace, they instead find a thoroughly modern city...
2019-07-03 18:31:00
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转载 Nash 嵌入定理
流形的度量改变意味着什么？ 1.首先来看最简单的例子： \(S^2:\) 将球面嵌入到 \(\mathbb{R}^3\)里面， 半径是 \(r\), 我们取标准球面坐标 \((\theta,\phi)\), 球面的度量是 \(ds^2=r^2（d\theta^2+sin\theta d\phi^2）\) ,改变度量 ，比如说 \(r\to 0\), 形状会缩小， 最终坍缩成为一个点。 ...
2018-08-26 18:38:00
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转载 Topology, Geometry, and Gauge Fields
\[\| F\|^2=\int_M (F_A,F_A)\] \[\boxed{ S = \int\nolimits_{t_1}^{t_2} \left\{ \int\nolimits_{\varSigma_t} N \left(R+K_{ij}K^{ij} -K^2\right) \sqrt{\gamma} \; \hbox{d}^3 x \right\} \hbox{d} t } \]...
2018-08-15 20:42:00
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转载 八月读书计划
科目 时间 学习形式 基础科目：Yang-Mills equation 周二，三，五，六（10：00~12：00&13：00~21：00） 讨论班 基础科目： FEM 周一，四，日（10：00~12：00& 13：00~21：00） 笔记，code 转载于:https://www.cnblogs.com/yuewen-chen/p/94775...
2018-08-14 20:15:00
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