Abstract
For difference equations with constant coefficients necessary and sufficient algebraic stability conditions are given for the stability definitions used by G. Forsythe and W. Wasow (A) and P. D. Lax and R. D. Richtmyer (B). The application of these conditions for difference equations with variable coefficients is considered and it is shown that the stability condition of definitionA is not sufficient for stability. The same is true with respect to the definitionB if the difference equations are not parabolic and do not approximate first order systems. Therefore another stability definition is proposed and a number of properties are discussed.
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Literatur
G. E. Forsythe and W. R. Wasow,Finite-Difference Methods For Partial Differential Equations. John Wiley & Sons, New York 1960.
P. D. Lax and R. D. Richtmyer,Survey of the stability of linear finite difference equations. Comm. Pure Appl. Math., 9 (1956), 267–293.
P. D. Lax,Differential equations, difference equations and matrix theory. Comm. Pure Appl. Math., 11 (1958), 175–194.
L. Gårding,Linear hyperbolic partial differential equations with constant coefficients. Acta Math., 85 (1951), 1–62.
V. S. Ryabenkii und A. F. Filippow,Über die Stabilität von Differenzengleichungen. Deutscher Verlag der Wissenschaften, Berlin 1960.
G. Dahlquist,Convergence and stability in the numerical integration of ordinary differential equations. Math. Scand., 4 (1956), 33–53.
F. John,On integration of parabolic equations by difference methods. Comm. Pure Appl. Math., 5 (1952), 155–211.
K. O. Friedrichs,Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math., 7 (1954), 345–392.
H. O. Kreiss, Über die Lösung des Cauchyproblems für lineare partielle Differentialgleichungen mit Hilfe von Differenzengleichungen. Acta Math., 101 (1959), 179–199.
H. O. Kreiss, Über die Differenzapproximation hoher Genauigkeit bei Anfangswertproblemen für partielle Differentialgleichungen. Num. Math., 1 (1959), 186–202.
P. D. Lax, On the stability of difference approximations to solutions of hyperbolic equations with variable coefficients. Comm. Pure Appl. Math., 14 (1961), 497–520.
P. D. Lax and B. Wendroff, On the stability of difference schemes. Manuskript.
I. Petrovskii, Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen im Gebiete der nichtanalytischen Funktionen. Bull. Univ. Etat. Moscou Ser. Int. Sect. A Fasc. 7 (1938), 1–74.
W. G. Strang,Difference methods for mixed boundary-value problems. Duke Math. Journal, 27 (1960), 221–232.
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Kreiss, HO. Über Die Stabilitätsdefinition Für Differenzengleichungen Die Partielle Differentialgleichungen Approximieren. BIT 2, 153–181 (1962). https://doi.org/10.1007/BF01957330
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DOI: https://doi.org/10.1007/BF01957330