Numerical methods for ordinary differential equations: Difference between revisions

Revision as of 01:55, 4 December 2023

Revision as of 10:25, 3 May 2024

Line 27:

== Methods ==

Numerical methods for solving first-order IVPs often fall into one of two large categories:<ref>Griffiths, D. F., & Higham, D. J. (2010). Numerical methods for ordinary differential equations: initial value problems. Springer Science & Business Media.</ref> [[linear multistep method]]s, or [[Runge–Kutta methods]]. A further division can be realized by dividing methods into those that are explicit and those that are implicit. For example, implicit [[linear multistep method]]s include [[Linear multistep method#Adams–Moulton methods|Adams-Moulton methods]], and [[Backward differentiation formula|backward differentiation methods]] (BDF), whereas [[implicit Runge–Kutta methods]]<ref>{{harvtxt|Hairer|Nørsett|Wanner|1993|pages=204–215}}</ref> include diagonally implicit Runge–Kutta (DIRK),<ref>Alexander, R. (1977). Diagonally implicit Runge–Kutta methods for stiff ODE’s. SIAM Journal on Numerical Analysis, 14(6), 1006-1021.</ref><ref>Cash, J. R. (1979). Diagonally implicit Runge-Kutta formulae with error estimates. IMA Journal of Applied Mathematics, 24(3), 293-301.</ref> singly diagonally implicit Runge–Kutta (SDIRK),<ref>Ferracina, L., & Spijker, M. N. (2008). Strong stability of singly-diagonally-implicit Runge–Kutta methods. Applied Numerical Mathematics, 58(11), 1675-1686.</ref> and Gauss–Radau<ref>Everhart, E. (1985). An efficient integrator that uses Gauss-Radau spacings. In International Astronomical Union Colloquium (Vol. 83, pp. 185-202185–202). Cambridge University Press.</ref> (based on [[Gaussian quadrature]]<ref>Weisstein, Eric W. "Gaussian Quadrature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GaussianQuadrature.html</ref>) numerical methods. Explicit examples from the [[Linear multistep method|linear multistep family]] include the [[Adams–Bashforth methods]], and any Runge–Kutta method with a lower diagonal [[Butcher tableau]] is [[explicit Runge–Kutta methods|explicit]]. A loose rule of thumb dictates that [[stiff equation|stiff]] differential equations require the use of implicit schemes, whereas non-stiff problems can be solved more efficiently with explicit schemes.

The so-called [[general linear methods]] (GLMs) are a generalization of the above two large classes of methods.<ref>Butcher, J. C. (1987). The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods. Wiley-Interscience.</ref>

Line 214:

== External links ==

* Joseph W. Rudmin, ''[http://csma31.csm.jmu.edu/physics/rudmin/ps.pdf Application of the Parker–Sochacki Method to Celestial Mechanics]'', 1998.

* Dominique Tournès, ''[https://web.archive.org/web/20130413090625/http://www.reunion.iufm.fr/dep/mathematiques/calculsavant/Equipe/tournes.html L'intégration approchée des équations différentielles ordinaires (1671-19141671–1914)]'', thèse de doctorat de l'université Paris 7 - Denis Diderot, juin 1996. Réimp. Villeneuve d'Ascq : Presses universitaires du Septentrion, 1997, 468 p. (Extensive online material on ODE numerical analysis history, for English-language material on the history of ODE numerical analysis, see, for example, the paper books by Chabert and Goldstine quoted by him.)

* {{cite journal

| last1=Pchelintsev | first1=A.N.