Consider the hyperbolic equation $u_t+sin(x)u_x-yu_y=0$ on the domain $[-\frac{\pi}{2},\frac{\pi}{2}]\times[-\frac{\pi}{2},\frac{\pi}{2}]$ with some initial condition $u(x,y,0)=\phi(x,y)$. Set up a convergent finite difference method and justify it.
My thought is to use a straightforward scheme that is backward time, central space. However, since this is not a linear PDE, the Lax equivalence theorem no longer applies. Is my scheme correct? If so, how could I fully justify it?