Solving PDE with state and time dependent boundary conditions

Question

I am interested in solving the following PDE (heat equation): $$\frac{\partial u}{\partial t} = \kappa \frac{\partial ^2 u}{\partial x^2}$$ In order to solve it, I discretize space uniformly into $N$ segments and convert the PDE into $N+1$ ODEs: $$ \frac{du_k}{dt} = \kappa\frac{u_{k+1} + u_{k-1} - 2u_k}{(\Delta x)^2} \quad k=0,1\ldots.N$$ which I can solve using any ODE routine (such as ode45 or ode15s in MATLAB). For the particular case of my problem, the Neumann boundary conditions for the PDE are dependent on the values of $u_k$ itself, i.e $$ \frac{du_b}{dx}= \left\{ \begin{array}{cc} \alpha & u_b < u_{critical}, \\ -\gamma u_b & u_b \geq u_{critical} \end{array} \right. $$ where $\alpha$ and $\gamma$ are constants and $b=\{0,N\}$.

Assuming that I am solving the above problem using some numerical ODE solver (such as ode45 in MATLAB), and there are two routines: odefun (which I provide to the solver) to evaluate the derivatives and odestep which I define and the solver calls it after every successful integration step, where should I check if the boundary condition needs to be changed, in odefun or odestep? I am asking this because the odefun routine might be used internally by the solver for evaluation of jacobian, errors etc and ideally, we should not change boundary conditions in the middle of an integration step, right?

Answer 1

You shouldn't have any problem coding in a conditional boundary condition into the odefun, like the one you defined in your problem statement.

Just use the last timestep's value of $u_b$ to do the conditional statement and grab the correct boundary condition value.

And the reality is, if your problem needs to change the boundary condition in the middle of integrating, then it has to do it. That's just how the physics are being modeled in your case. Given that these dynamics are okay, you should ideally incorporate that boundary condition change whenever you need to.