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
\[\begin{align}
u_t &=(\frac{u_x}{\sqrt{1+u_x^2}})_x+S(x)\\
u_x&=\frac{1}{\varepsilon^2+(x-x_0)^2} \in \{ x_L,x_R\}\\
\end{align}
\]
where
\[S(x)=-\frac{2 \text{x0}-2 x}{\sqrt{\frac{1}{\left(\text{eps}^2+(x-\text{x0})^2\right)^2}+1} \left(\text{eps}^4+2 \text{eps}^2 (x-\text{x0})^2+x^4-4 x^3 \text{x0}+6 x^2 \text{x0}^2-4 x \text{x0}^3+\text{x0}^4+1\right)}
\]
- 测试表明，WENO5 使用 Guangshan-Jang WENO5 在边界处的插值是略有问题的，继续测试原版的 WENO5_Shu
- 测试IMEX-WENO5
\[\begin{align}
u_t &= (\frac{u_x}{\sqrt{1+u_x^2}})_x+S(x)\\
\frac{1}{dt}u^{n+1}-\frac{1}{dt}u^n &=J+S+\Delta u^{n+1}-\Delta u^n\\
\frac{1}{dt}u^{n+1}-\Delta u^{n+1} &=\frac{1}{dt}u^n+J(u^n)+S -\Delta u^n\\
(\frac{1}{dt}I-\Delta )u^{n+1} &= \frac{1}{dt}u^n+J(u^n)+S -\Delta u^n\\
u^{n+1} &=L^{-1}(\frac{1}{dt}u^n+J(u^n)+S -\Delta u^n)
\end{align}
\]
