Numerical methods for ordinary differential equations: Difference between revisions

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== 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>

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===First-order exponential integrator method===

{{details|Exponential integrator}}

Exponential integrators describe a large class of integrators that have recently seen a lot of development.<ref name="Exponential integrators">{{harvtxt|Hochbruck|Ostermann|2010|pp=209–286}} This is a modern and extensive review paper for exponential integrators</ref> They date back to at least the 1960s.

In place of ({{EquationNote|1}}), we assume the differential equation is either of the form

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*{{cite book|last=Bradie|first=Brian|title=A Friendly Introduction to Numerical Analysis|year=2006|publisher=Pearson Prentice Hall|___location=Upper Saddle River, New Jersey|isbn=978-0-13-013054-9}}

*[[John C. Butcher|J. C. Butcher]], ''Numerical methods for ordinary differential equations'', {{isbn|0-471-96758-0}}

*{{cite book

*Ernst Hairer, Syvert Paul Nørsett and Gerhard Wanner, ''Solving ordinary differential equations I: Nonstiff problems,'' second edition, Springer Verlag, Berlin, 1993. {{isbn|3-540-56670-8}}.

| last1 = Hairer | first1 = E.

| last2 = Nørsett | first2 = S. P.

| last3 = Wanner | first3 = G.

| edition = 2nd

| isbn = 3-540-56670-8

| mr = 1227985

| publisher = Springer-Verlag, Berlin

| series = Springer Series in Computational Mathematics

| title = Solving Ordinary Differential Equations. I. Nonstiff Problems

| volume = 8

| year = 1993}}

*Ernst Hairer and Gerhard Wanner, ''Solving ordinary differential equations II: Stiff and differential-algebraic problems,'' second edition, Springer Verlag, Berlin, 1996. {{isbn|3-540-60452-9}}. <br> ''(This two-volume monograph systematically covers all aspects of the field.)''

*{{cite journal|last=Hochbruck|first=Marlis|author1-link=Marlis Hochbruck|author2-last=Ostermann, |author2-first=Alexander |title=Exponential integrators|journal=Acta Numerica|volume=19|date=May 2010|pages=209–286|doi=10.1017/S0962492910000048|bibcode=2010AcNum..19..209H|citeseerx=10.1.1.187.6794|s2cid=4841957 }}

*Arieh Iserles, ''A First Course in the Numerical Analysis of Differential Equations,'' Cambridge University Press, 1996. {{isbn|0-521-55376-8}} (hardback), {{isbn|0-521-55655-4}} (paperback). <br> ''(Textbook, targeting advanced undergraduate and postgraduate students in mathematics, which also discusses [[numerical partial differential equations]].)''

*John Denholm Lambert, ''Numerical Methods for Ordinary Differential Systems,'' John Wiley & Sons, Chichester, 1991. {{isbn|0-471-92990-5}}. <br> ''(Textbook, slightly more demanding than the book by Iserles.)''

== External links ==

* Joseph W. Rudmin, ''[http://csma31.csm.jmu.edu/physics/rudmin/ps.pdf Application of the Parker–Sochacki Method to Celestial Mechanics] {{Webarchive|url=http://arquivo.pt/wayback/20160516155343/http://csma31.csm.jmu.edu/physics/rudmin/ps.pdf |date=2016-05-16 }}'', 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.