Domain of dependence

The domain of dependence of a solution value consists of the initial and boundary locations whose data can influence it. For $u_t+a{u}_x=0$ on the whole line, $u(x,t)=u(x-at,0)$, so its initial-data domain of dependence is the foot of the characteristic. For a finite-difference scheme, repeated stencil dependencies determine a numerical domain of dependence. The Courant–Friedrichs–Lewy condition requires the numerical domain to cover the physical one for convergence; it is necessary under its usual hyperbolic hypotheses, but not in general sufficient for stability.