Heat equation with Neumann and Dirichlet conditions on same boundary

Question

I am looking at numerical solutions to the heat equation with Dirichlet and Neumann conditions on the same boundary. That is $u(x,t)$ satisfying $$ u_t = u_{xx}\,, \quad x \in[0,1]\,, \quad t>0\,, $$ with $$ \begin{gathered} u(x,0) = u_0(x)\,,\\ u(0,t) = g(t) \quad \forall \, t>0\,,\\ u_x(0,t) = h(t) \quad \forall \, t>0. \end{gathered} $$ This is with a view to solving an advection-diffusion problem with the same BCs.

If I discretise on $N+1$ points, $$ U^n_j \approx u(j \Delta x, n \Delta t) $$ with $\Delta x = 1/N$ then I would expect to discretise as $$ (U_j)_t = \frac{1}{\Delta x^2}\left[ U_{j-1} - 2 U_j + U_{j+1} \right]\,, $$ but then I can't see how to discretise both the boundary conditions. If I let $$ U_0(t) = g(t)\,, $$ and $$ \frac{U_1(t) - U_0(t)}{\Delta x} = h(t) $$ then I have an issue determining $(U_N)_t$.

Resentfully I tried discretising with backwards difference twice $$ (U_j)_t = \frac{1}{\Delta x^2}\left[ U_{j-2} - 2 U_{j-1} + U_{j} \right]\,, $$ which allows me to impose both boundary conditions on $(U_2)_t$, but I think this is unconditionally unstable.

Does this even make sense? I was thinking about a shooting method using a shooting parameter $s$ with $u(1,t)=s$ to try to match $u_x(0,t;s)=h(t;s)$ at each time step, but this feels inefficient.

Answer 1

In the comments Christian directed me towards lateral Cauchy problems and the fact that this is a textbook example of an ill-posed problem.

Following this lead, I found that this is more specifically know as the sideways heat equation. As an alternative to the suggested quasireversibility method (again Christian), there is a proposed sequential solution in Berntsson (2003).

[Berntsson (2003)] Fredrik Berntsson , Sequential solution of the sideways heat equation by windowing of the data, 2003, Inverse Problems in Engineering, (11), 2, 91-103.