Numerical method for solving PDE with non-linear boundary conditions

Question

I have the following problem:

\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}

Where \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}

Initial and boundary conditions are:

\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}

The question is what numerical method is suitable to solve this equations?

What I tried is to solve this equation using Finite difference method but there is non-linear boundary condition, that does not allow to use that method.

Answer 1

All right, so $dr_{d}/dt$ is coupled to $w$ because your diffusion coefficient is a function of $w$. The boundary condition being nonlinear in $w$ isn't a big deal, because after discretization in space, you'll have a collection of differential and algebraic equations in space, and you can use an implicit method to handle it. The trickier thing is that $r_{d}$ changes with time, because that affects how you handle the spatial discretization.

As a first attempt, I'd rescale the spatial coordinate $r$, and re-express your PDE for $w$ in terms of $\xi = r/r_{d}$. In $\xi$, your BCs are now at $0$ and $1$, and it shouldn't really impact the radial symmetry BC or the Dirichlet initial condition. It will affect the flux BC at $r = r_{d}$ (now $\xi = 1$). Rescaling the radial coordinate makes it possible to discretize $\xi$ on $[0,1]$ with fixed node locations (in $r$, the node locations will change with time, except for the node at 0). You should be able to use a finite-difference discretization, and it will be fine.

Other approaches are possible, like FEM with cut cells, but I wouldn't investigate that sort of approach right away unless you're familiar with it.