Are spatial boundary conditions required for PDEs discretized with Method of Lines?

Let's take the example of the unsteady one-dimensional heat equation inside a solid on a domain $xin[0,1]$: $$partial_t u - Dpartial_{xx} u = 0$$ with the initial profil $u(0,x) = u_0(x)$ at $t=0$, and Dirchlet boundary conditions enforcing that the wall at $x=0$ (respectively $x=1$) is at temperature $u_{L}$ (respectively $u_{R}$). If we use discretise our solid with a uniformely distributed set of points $i in [1,N]$ at positions $x_i = i Delta x$ with $Delta x = 1/(N+1)$ and use second order centered finite differences for the diffusion term, we obtain the following semi-discrete set of ODEs: $$d_t u_i = Dfrac{u_{i+1} - 2u_i + u_{i-1}}{Delta x^2}$$ for $i in [2, N-1]$. The boundary points $i=1$ and $i=N$ must be taken care of separately to ensure the boundary conditions are respected. Let's take the example of the boundary at $x=0$. The temperature $u(x=0)$ is enforced to be 0. Our first point is as $x=1/(n+1)$, therefore we can write: $$d_t u_1 = Dfrac{u_{2} - 2u_1 + u_{L}}{Delta x^2}$$

Similarly for our last point: $$d_t u_N = Dfrac{u_{N-1} - 2u_N + u_{R}}{Delta x^2}$$

We have to use these formulations, otherwise we would not even be able to compute the centered finite difference approximation of the diffusion terms at these outer points. An error would for example be to aproximate these terms with a decentered fintie difference formula, e.g for the point $i=1$: $$ d_t u_1 = Dfrac{ 2u_1 - 5u_2 + 4 u_3 - u_4}{Delta x^2} $$ which uses a second-order forward finite difference expression of the diffusion term. But then, there is not guarantee that the temperature at $x=0$ will remain at $u_L$, i.e. your system will most likely have an unphysical behaviour.

Sometimes we rather use a Neumann boudary condition to specify a given heat flux at the boundary. Let's take the case of a Neumann BC at $x=0$, with a zero heat flux, i.e. $frac{partial u}{partial x}(x=0) = 0$. This condition can be discretised as $frac{u_1-u_L}{Delta x}=0$, which means that $u_L(t) = u_1(t)$ at all times $t$. You can then inject this value of $u_L$ in the second equation of this answer.

Technically, a discretized PDE is integrated in time as follows:

• the time integration routines gives you the discrete solution profile $u_i, i in [1,N]$ at time $t_n$
• we compute the boundary values $u_L$ and $u_R$ so as to enforce the proper boundary conditions
• we compute the time derivatives of our discrete solution values and return them to time integration routine.

I don't know how you've managed to integrate your system without specifying BCs at all times, maybe its just luck in your formulation or the particular equation you consider. But in the case of the discrete heat equation I've treated here, I see 3 potential ways of having this problem:

• you forget to specify $d_t u_1$ and then you actually enforce a Dirichlet boundary condition that reads $u(t,x_1)= u_0(x_1)$, i.e. the wall temperature remains equal to its initial value

• you forget the term $u_L$ in the third equation, which the means you implicitly assume $u_L=0$ .

• you used the forward finite difference expression, thus omitting the BC value. In this case, we can see what the implcitly asusmed of $u_L$ is by equating the 3rd and 5th equation and we obtain: $u_L = -4u_1 + 6u_2 - 4u_3 + u_4$, that is $u_L$ is relatively "unpredictable", and consequently so will be $u_1$, and $u_2$ and so on...