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

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

Comments

The Courant–Friedrichs–Lewy condition is essential for the convergence and stability of explicit difference schemes for hyperbolic equations cf. [a1]–[a5]. Reference [a2] is the translation of [1] into English.

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)
[a1]	R. Courant, K.O. Friedrichs, "Supersonic flow and shock waves" , Interscience (1948)
[a2]	R. Courant, K.O. Friedrichs, H. Lewy, "On the partial difference equations of mathematical physics" , NYO-7689 , Inst. Math. Sci. New York Univ. (1956) (Translated from German)
[a3]	G.E. Forsythe, W.R. Wasow, "Finite difference methods for partial differential equations" , Wiley (1960)
[a4]	A.R. Mitchell, D.F. Griffiths, "The finite difference method in partial equations" , Wiley (1980)
[a5]	R.D. Richtmeyer, K.W. Morton, "Difference methods for initial value problems" , Wiley (1967)