Abstract
The analysis presented in this chapter evolved soon after 1950, when discretisations of hyperbolic and parabolic differential equations had to be developed. Most of the material can be found in Richtmyer–Morton [21], see also Lax–Richtmyer [16]. All results concern linear differential equations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Courant, R., Friedrichs, K.O., Lewy, H.: Über die partiellen Differentialgleichungen der mathematischen Physik. Math. Ann. 100, 32–74 (1928)
Crank, J., Nicolson, P.: A practical method for numerical evaluation of partial differential equations of the heat-conduction type. Proc. Cambridge Phil. Soc. 43, 50–67 (1947). (reprint in Adv. Comput. Math. 6, 207–226, 1996)
de Moura, C.A., Kubrusly, C.S. (eds.): The Courant–Friedrichs–Lewy (CFL) condition. Springer, New York (2013)
Du Fort, E.C., Frankel, S.P.: Stability conditions in the numerical treatment of parabolic differential equations. Math. Tables and Other Aids to Computation 7, 135–152 (1953)
Friedrichs, K.O.: Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math. 7, 345–392 (1954)
Hackbusch, W.: Iterative Lösung großer schwachbesetzter Gleichungssysteme, 2nd ed. Teubner, Stuttgart (1993)
Hackbusch, W.: Iterative solution of large sparse systems of equations. Springer, New York (1994)
Hackbusch, W.: Elliptic differential equations. Theory and numerical treatment, Springer Series in Computational Mathematics, vol. 18, 2nd ed. Springer, Berlin (2003)
Kreiss, H.O.: Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren. BIT 2, 153–181 (1962)
Kreiss, H.O.: On difference approximations of the dissipative type for hyperbolic differential equations. Comm. Pure Appl. Math. 17, 335–353 (1964)
Kreiss, H.O.: Difference approximations for the initial-boundary value problem for hyperbolic differential equations. In: D. Greenspan (ed.) Numerical solution of nonlinear differential equations, pp. 141–166. Wiley, New York (1966)
Kreiss, H.O.: Initial boundary value problem for hyperbolic systems. Comm. Pure Appl. Math. 23, 277–298 (1970)
Kreiss, H.O., Wu, L.: On the stability definition of difference approximations for the initial boundary value problem. Appl. Numer. Math. 12, 213–227 (1993)
Kröner, D.: Numerical Schemes for Conservation Laws. J. Wiley und Teubner, Stuttgart (1997)
Lax, P.D.: Difference approximations of linear differential equations – an operator theoretical approach. In: N. Aronszajn, C.B. Morrey Jr (eds.) Lecture series of the symposium on partial differential equations, pp. 33–66. Dept. Math., Univ. of Kansas (1957)
Lax, P.D., Richtmyer, R.D.: Survey of the stability of linear finite difference equations. Comm. Pure Appl. Math. 9, 267–293 (1965)
Lax, P.D., Wendroff, B.: Systems of conservation laws. Comm. Pure Appl. Math. 13, 217–237 (1960)
Lax, P.D., Wendroff, B.: Difference schemes for hyperbolic equations with high order of accuracy. Comm. Pure Appl. Math. 17, 381–398 (1964)
LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)
Mizohata, S.: The theory of partial differential equations. University Press, Cambridge (1973)
Richtmyer, R.D., Morton, K.W.: Difference Methods for Initial-value Problems, 2nd ed. John Wiley & Sons, New York (1967). Reprint by Krieger Publ. Comp., Malabar, Florida, 1994
Riesz, F., Sz.-Nagy, B.: Functional Analysis. Dover Publ. Inc, New York (1990)
Riesz, M.: Sur les maxima des formes bilinéaires et sur les fonctionelles linéaires. Acta Math. 49, 465–497 (1926)
Thomée, V.: Stability of difference schemes in the maximum-norm. J. Differential Equations 1, 273–292 (1965)
Thorin, O.: An extension of the convexity theorem of M. Riesz. Kungl. Fysiogr. Sällsk. i Lund Förh. 8(14) (1939)
Tomoeda, K.: Stability of Friedrichs’s scheme in the maximum norm for hyperbolic systems in one space dimension. Appl. Math. Comput. 7, 313–320 (1980)
Zeidler, E. (ed.): Oxford Users’ Guide to Mathematics. Oxford University Press, Oxford (2004)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Hackbusch, W. (2014). Instationary Partial Differential Equations. In: The Concept of Stability in Numerical Mathematics. Springer Series in Computational Mathematics, vol 45. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39386-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-39386-0_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39385-3
Online ISBN: 978-3-642-39386-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)