Content deleted Content added
ParallelBot (talk | contribs) Tags: references removed Visual edit |
→References: templatize ref to fix harv linking error |
||
(8 intermediate revisions by 6 users not shown) | |||
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.
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 60:
===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
Line 110:
Some IVPs require integration at such high temporal resolution and/or over such long time intervals that classical serial time-stepping methods become computationally infeasible to run in real-time (e.g. IVPs in numerical weather prediction, plasma modelling, and molecular dynamics). [[Parallel-in-time]] (PinT) methods have been developed in response to these issues in order to reduce simulation runtimes through the use of [[parallel computing]].
Early PinT methods (the earliest being proposed in the 1960s)<ref> {{cite journal |last=Nievergelt |first=Jürg |date=1964 |title=Parallel methods for integrating ordinary differential equations |journal=Communications of the ACM |volume=7 |issue=12 |pages=731–733 |doi=10.1145/355588.365137 |s2cid=6361754 |doi-access=free}}</ref>
▲{{cite journal |last=Nievergelt |first=Jürg |date=1964 |title=Parallel methods for integrating ordinary differential equations |journal=Communications of the ACM |volume=7 |issue=12 |pages=731–733 |doi=10.1145/355588.365137 |s2cid=6361754 |doi-access=free}}</ref>) were initially overlooked by researchers due to the fact that the parallel computing architectures that they required were not yet widely available. With more computing power available, interest was renewed in the early 2000s with the development of [[Parareal]], a flexible, easy-to-use PinT algorithm that is suitable for solving a wide variety of IVPs. The advent of [[exascale computing]] has meant that PinT algorithms are attracting increasing research attention and are being developed in such a way that they can harness the world's most powerful [[Supercomputer|supercomputers]]. The most popular methods as of 2023 include Parareal, PFASST, ParaDiag, and MGRIT<ref>{{Cite web |title=Parallel-in-Time.org |url=https://parallel-in-time.org/methods/index.html |url-status=live |access-date=15/11/23 |website=Parallel-in-Time.org}}</ref>.
== Analysis ==
Line 192 ⟶ 190:
: <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
== See also ==
Line 208 ⟶ 206:
*{{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
| 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
*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 (
* {{cite journal
| last1=Pchelintsev | first1=A.N.
|