[Math] Finite Difference without boundary conditions

finite differencesnumerical methods

I'm working through the paper where the Finite Difference method is employed to solve the PDE

$\displaystyle \frac{\partial u(x,t)}{\partial t} = a \cdot \frac{\partial^2 u(x,t)}{\partial x^2} + b \cdot \frac{\partial u(x,t)} {\partial x} + c \cdot u(t,x) $

on $\mathbb{R}^{+} \times [0, T]$, but corresponding boundary conditions aren't known. Introducing the $x$-mesh on $[0, B]$, $x_i = i \cdot h$, where $h = \dfrac{B}{m}$, $i \in \{0, …, m\}, m \in \mathbb{N}$, derivatives at points $x = x_0$, $x = x_m$ are thus approximated using forward and backward differences for each time level, respectively (e.g. $\dfrac{\partial u}{\partial x_o} \approx \dfrac{u_1 – u_0}{h}$ and so on). The resulting system matrix is thus of size $(m+1) \times (m+1)$.

My question is which stability/convergence results do we get here compared to the case with boundary conditions imposed? When implementing the method, I encountered an issue that values of approximated solution with $x$ close to $B$ vary heavily for different $B$'s. For example, taking $B= 40.4$ and $B = 22000$ I get:

approximation for different B's

I wonder whether this an implementation issue or consequence of no boundary.

Best Answer

Usually, when no boundary conditions are provided, this kind of problem has not a unique solution. There are some cases where a "no boundary condition" is used, usually for only one part of the boundary of the domain where an outflow condition is required, but these cases are rare and this technique still no well understood AFAIK.

Did you check whether the matrix of your linear system is singular or not?

If it is singular (or nearly), this could explain the high sensibility that you see.

Another possibility is that you impose something without really noticing it. You should show the exact equations that you use for the discretization, in particular how you treat the case close to the boundary of the domain.

EDIT: After your precision, I can see two points that can make your solution depend on $B$:

  • I'm still not sure whether the continuous problem has a unique solution without boundary conditions. I know some cases (e.g. Poisson problem on a square with one "open" side) where the continuous problem without boundary conditions is ill-posed while the discrete problem yields a non-singular linear system to solve (in a FEM context). So, you can get a discrete solution, but you do not know which solution is approximated. Different $B$ can then yield different solutions.
  • "No boundary condition" means "I don't know at all what is after this point", however, when you increase your domain size, you actually give more information to your solution, since you provide an initial value for the solution after the point $x=44$, and this information will change the solution in the domain $(0,44)$ due to the diffusion.

The fact that the solution differs more on the right side than on the left side seems to indicate that the second point is the reason that you were looking for. To confirm this, try to decrease the diffusivity $a$ or increase the transport ($b>0$) and see if the two solutions get closer.

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