SAT Boundary Conditions

Imagine two characteristic grid functions $U^\pm(x,t)$ on a grid with $n$ points and a domain $x\in [a,b]$. These grid functions have advection speeds of $c_\pm$ that are positive and negative respectively. We use SBP operators for spatial derivatives, with norm $\Sigma_{ij}$. We would like to apply the boundary conditions $U(a,t)^+=f(t)$ and $U^-(b,t)=g(t)$ weakly using SATs, with strengths $s_1$ and $s_n$ to be determined

$$ \partial_t U^+(a,t)= \cdots + \frac{s_1}{\Sigma_{11}}\left[f(t)-U^+(a,t)\right],,\quad\partial_t U^-(b,t)= \cdots + \frac{s_n}{\Sigma_{nn}}\left[g(t)-U^-(b,t)\right],. $$

You can show (see below) that for proper SBP energy conservation the strengths must be equal to the magnitude of the incoming advection speeds. At the left boundary we have a positive speed $s_1\rightarrow c_+$ and at the right boundary we have a negative speed $s_n\rightarrow -c_-$. In the case of the wave equation around a static background black hole in Kerr-Schild coordinates, at the right boundary we have $s_n=1$, but at the left boundary we have $s_1<1$, which goes to zero in the limit the boundary gets close to the horizon. Interestingly, it seems the weakness of the SAT is coupled to the time-scale that characteristics are exchanged at the boundary.

Proved in this paper: https://global-sci.org/intro/article_detail/cicp/7621.html