I am studying the spherically symmetric wave equation on the Schwarzschild background in Eddington-Finkelstein coordinates $(v,r)$. I want to numerically integrate the $v$-time-evolution of such a wave. However, I realised that I seem to misunderstand something, because the following problem arises:
For integration of the wave equation, I apply a Finite-Difference-Scheme to the spatial derivatives. In order for the simulation to be numerically stable, the grid sizes $(\Delta v,\Delta r)$ must satisfy the Courant-condition
$$\frac{\Delta r}{\Delta v} > \frac{u}{C}$$
where $C < 1$ is the Courant-factor and $u$ is the speed of the wave. Eddington Finkelstein coordinates are constructed such that radially ingoing null geodesics satisfy $v = \text{const}$. If this is the case, then -- when integrating the differential equation -- in order to obtain the solution for the next time step $(v_0 + \Delta v)$ at some point $r = r_0$, each spatial point $r > r_0$ must be considered, since they all may influence the solution at $(v_0 + \Delta v, r_0)$. For a usual number of spatial gridpoints, this would require an asymmetric FDA-Scheme of an order much too high to be practical.
What is the flaw in my thinking here?
