Consider the following problem $$ W_{uv} = F $$ where the forcing term can depend on $u,v$ (see Edit 1 below for the formulation), and $W$ and its first derivatives. This is a 1+1 dimensional wave equation. We have initial data prescribed at $\{u+v = 0\}$.
I am interested in the solution inside the domain of dependence of an interval $$\{ u+v = 0, u \in [- u_M,u_M]\}$$ and am considering the following finite difference scheme.
- The goal is to evolve $W_u$ by $W_u(u,v+1) - W_u(u,v) = F(u,v)$ and similarly $W_v(u+1,v) - W_v(u,v) = F(u,v)$. This scheme is integrable in the sense that $$ W(u,v) + W_u(u,v) + W_v(u+1,v) = W(u+1,v+1) = W(u,v) + W_v(u,v) + W_u(u,v+1)$$ so I can consistently compute $W$ from the initial data by integrating upwards; hence I only really need to look at the evolution equations for $W_v$ and $W_u$.
- For the initial data, we need the compatibility condition $W_u(u,v) - W_v(u+1,v-1) = W(u+1,v-1) - W(u,v)$. Which suggests that I can compute the initial data by using the forward (in $u$) finite difference of $W$ on the initial time with the values of given $W_t$ at half-integer points $(u+0.5,v-0.5)$.
Question:
- Is this a well known scheme? In particular, where can I find analysis of this scheme?
- Any thing obvious I should look out for?
Background: Pretend I know next to nothing (which is probably true, as I am a pure mathematician trying to learn a little bit of computation machinery).
Edit 1: Just to clarify (to address some comments): the equation in $x$ $t$ coordinates would be $$ W_{tt} - W_{xx} = F $$ and $u$ and $v$ are ¨null coordinates¨ given by (up to some renormalising factors of 2) $u = t+x$ and $v = t-x$. So the initial data at $\{u+v = 0\}$ is in fact at $\{t = 0\}$.
So instead of a mesh adapted to $(t,x)$ I consider a mesh adapted to $(u,v)$ which is ¨rotated 45 degrees¨. Compared to the $(t,x)$ where $t,x$ take integer values, one can think of the $u,v$ mesh as having additional points where both (but not just one of) $t$ and $x$ take half-integer values.