View source for Courant-Friedrichs-Lewy condition

A necessary condition for the stability of difference schemes in the class of infinitely-differentiable coefficients. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026760/c0267601.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026760/c0267602.png" /> be the dependence region of the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026760/c0267603.png" /> of the solution to the corresponding difference equation. A necessary condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026760/c0267604.png" /> to be convergent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026760/c0267605.png" /> is that, as the grid spacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026760/c0267606.png" /> is diminished, the dependence region of the difference equation covers the dependence region of the differential equation, in the sense that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026760/c0267607.png" /></td> </tr></table>

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



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Courant,   K.O. Friedrichs,   "Supersonic flow and shock waves" , Interscience  (1948)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.E. Forsythe,   W.R. Wasow,   "Finite difference methods for partial differential equations" , Wiley  (1960)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.R. Mitchell,   D.F. Griffiths,   "The finite difference method in partial equations" , Wiley  (1980)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.D. Richtmeyer,   K.W. Morton,   "Difference methods for initial value problems" , Wiley  (1967)</TD></TR></table>