计算科学 - 非线性边界条件下求解偏微分方程的数值方法 - 吾爱随笔录

我有以下问题：

\begin{aligned} \frac{\partial w}{\partial t} = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} D_{1} \frac{\partial w}{\partial r}) \\ \frac{d r_{d}}{d t} = C_{1} \frac{(C_{2} + C_{3} {(\frac{r_{d}}{r_{d, 0}})}^{0.5}) D_{1}}{2 r_{d}} \end{aligned}

$\begin{align} \frac{\partial w}{\partial t} = \frac{1}{r^2} \frac{\partial}{\partial r}\left( r^2 D_1 \frac{\partial w}{\partial r} \right) \\ \\ \frac{dr_d}{dt} = C_1\frac{\left(C_2 + C_3 \left( \frac{r_d}{r_{d,0}}\right)^{0.5} \right) D_1}{2r_d} \end{align}$

其中

\begin{aligned} D_{1} & = A e^{B w} \\ A & = const \\ B & = const \\ C_{1} & = const \\ C_{2} & = const \\ C_{3} & = const \\ r_{d, 0} & = const \\ t & \geq 0 \\ r & \in [0, r_{d}] \end{aligned}

$\begin{align} D_1 &= A e^{B w} \\ A &=\operatorname{const} \\ B &=\operatorname{const} \\ C_1 &=\operatorname{const} \\ C_2 &=\operatorname{const} \\ C_3 &=\operatorname{const} \\ r_{d, 0} &=\operatorname{const} \\ t &\ge 0 \\ r &\in [0, r_d] \end{align}$

初始条件和边界条件为：

\begin{aligned} w (0, r) & = w_{0} \\ \frac{\partial w (t, 0)}{\partial r} & = 0 \\ D_{1} \frac{\partial w (t, r_{d})}{\partial r} & = (1 - w)_{r_{d}} \frac{d r_{d}}{d t} \end{aligned}

$\begin{align} w(0, r) &= w_0 \\ \frac{\partial w(t, 0)}{\partial r} &= 0 \\ D_1\frac{\partial w(t, r_d)}{\partial r} &= (1-w)_{r_d} \frac{dr_d}{dt} \end{align}$

问题是什么数值方法适合求解这个方程？

我尝试的是使用有限差分法求解该方程，但存在非线性边界条件，不允许使用该方法。