Courant-Friedrichs-Lewy condition

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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



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.


[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)
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