# Courant-Friedrichs-Lewy condition

A necessary condition for the stability of difference schemes in the class of infinitely-differentiable coefficients. Let $\Omega(P)$ be the dependence region for the value of the solution with respect to one of the coefficients (in particular, the latter might be an initial condition) and let $\Omega_h(P)$ be the dependence region of the value $u_h(P)$ of the solution to the corresponding difference equation. A necessary condition for $u_h(P)$ to be convergent to $u(P)$ is that, as the grid spacing $h$ is diminished, the dependence region of the difference equation covers the dependence region of the differential equation, in the sense that

$$\Omega(P)\subset\varlimsup_{h\to0}\Omega_h(P).$$

#### References

 [1] R. Courant, K.O. Friedrichs, H. Lewy, "Ueber die partiellen Differenzgleichungen der mathematische Physik" Math Ann. , 100 (1928) pp. 32–74 [2] S.K. Godunov, V.S. Ryaben'kii, "The theory of difference schemes" , North-Holland (1964) (Translated from Russian)