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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: <math> \frac{u_{i+1}-2u_{i}+u_{i-1}}{h^2}-u_i = 0, \quad \forall i={1,2,3,...,n-1}.</math>

On first viewing, this system of equations appears to have difficulty associated with the fact that the equation involves no terms that are not multiplied by variables, but in fact this is false. At ''i'' = 1 and ''n'' − 1 there is a term involving the boundary values <math>u(0)=u_0 </math> and <math> u(1)=u_n </math> and since these two values are known, one can simply substitute them into this equation and as a result have a non-homogeneous [[System_of_linear_equations|linear system of linear equations]] that has non-trivial solutions.

== See also ==

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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.