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Difference between revisions of "Courant-Friedrichs-Lewy condition"

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$$\Omega(P)\subset\varlimsup_{h\to0}\Omega_h(P).$$
 
$$\Omega(P)\subset\varlimsup_{h\to0}\Omega_h(P).$$
 
====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====
 
====Comments====
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====References====
 
====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>
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<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>
 +
<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>

Latest revision as of 07:04, 27 May 2023

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)
How to Cite This Entry:
Courant-Friedrichs-Lewy condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Courant-Friedrichs-Lewy_condition&oldid=53954
This article was adapted from an original article by N.S. Bakhvalov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article