View source for Numerical methods for ordinary differential equations

{{Short description|Methods used to find numerical solutions of ordinary differential equations}}
[[File:Numerical integration illustration, step=1.svg|right|180px|thumb|Illustration of numerical integration for the differential equation <math>y'=y, y(0)=1.</math>
{{legend|#0000F0|Blue: Euler method}}
{{legend|#00D000|Green: [[Midpoint method]]}}
{{legend|#F00000|Red: Exact solution: {{nowrap|<math display="inline">y=e^t</math>}}.}}
The step size is {{nowrap|<math>h=1.0</math>}}.]]
[[File:Numerical integration illustration step=0.25.svg|right|180px|thumb|The same illustration for <math>h=0.25.</math> The midpoint method converges faster than the Euler method, as <math>h \to 0</math>.]]

'''Numerical methods for ordinary differential equations''' are methods used to find [[Numerical analysis|numerical]] approximations to the solutions of [[ordinary differential equation]]s (ODEs). Their use is also known as "[[numerical integration]]", although this term can also refer to the computation of [[integral]]s.

Many differential equations cannot be solved exactly. For practical purposes, however&nbsp;– such as in engineering&nbsp;– a numeric approximation to the solution is often sufficient. The [[algorithm]]s studied here can be used to compute such an approximation. An alternative method is to use techniques from [[calculus]] to obtain a [[series expansion]] of the solution.

Ordinary differential equations occur in many scientific disciplines, including [[physics]], [[chemistry]], [[biology]], and [[economics]].<ref>Chicone, C. (2006). Ordinary differential equations with applications (Vol. 34). Springer Science & Business Media.</ref> In addition, some methods in [[numerical partial differential equations]] convert the [[partial differential equation]] into an ordinary differential equation, which must then be solved.

== The problem ==

A first-order differential equation is an [[Initial value problem]] (IVP) of the form,<ref>{{harvtxt|Bradie|2006|pp=533–655}}</ref>
{{NumBlk|:|<math>y'(t) = f(t,y(t)), \qquad y(t_0)=y_0, </math>|{{EquationRef|1}}}}
where <math>f</math> is a function <math>f:[t_0, \infty) \times \R^d \to \R^d</math>, and the initial condition <math>y_0 \in \R^d </math> is a given vector.  ''First-order'' means that only the first derivative of ''y'' appears in the equation, and higher derivatives are absent.

Without loss of generality to higher-order systems, we restrict ourselves to ''first-order'' differential equations, because a higher-order ODE can be converted into a larger system of first-order equations by introducing extra variables.  For example, the second-order equation {{nowrap|1=''y''<nowiki>′′</nowiki> = −''y''}} can be rewritten as two first-order equations: {{nowrap|1=''y''<nowiki>′</nowiki> = ''z''}} and {{nowrap|1=''z''′ = −''y''.}}

In this section, we describe numerical methods for IVPs, and remark that ''[[boundary value problem]]s'' (BVPs) require a different set of tools.  In a BVP, one defines values, or components of the solution ''y'' at more than one point.  Because of this, different methods need to be used to solve BVPs.  For example, the [[shooting method]] (and its variants) or global methods like [[finite difference]]s,<ref name="fdm">LeVeque, R. J. (2007). Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems (Vol. 98). SIAM.</ref> [[Galerkin method]]s,<ref>Slimane Adjerid and Mahboub Baccouch (2010) Galerkin methods. Scholarpedia, 5(10):10056.</ref> or [[collocation method]]s are appropriate for that class of problems.

The [[Picard&ndash;Lindelöf theorem]] states that there is a unique solution, provided ''f'' is [[Lipschitz continuity|Lipschitz-continuous]].

== 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. 185–202). 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>

===Euler method===
<!--Linked from [[Trigonometric tables]]-->
{{details|Euler method}}
From any point on a curve, you can find an approximation of a nearby point on the curve by moving a short distance along a line [[tangent]] to the curve.

Starting with the differential equation ({{EquationNote|1}}), we replace the derivative ''y''′ by the [[finite difference]] approximation
{{NumBlk|:|<math> y'(t) \approx \frac{y(t+h) - y(t)}{h}, </math>|{{EquationRef|2}}}}
which when re-arranged yields the following formula
:<math> y(t+h) \approx y(t) + hy'(t) </math>
and using ({{EquationNote|1}}) gives:
{{NumBlk|:|<math> y(t+h) \approx y(t) + hf(t,y(t)). </math>|{{EquationRef|3}}}}
This formula is usually applied in the following way. We choose a step size ''h'', and we construct the sequence <math>t_0, t_1 = t_0 + h, t_2 = t_0 + 2h,...</math>   We denote by <math>y_n</math> a numerical estimate of the exact solution <math>y(t_n)</math>. Motivated by ({{EquationNote|3}}), we compute these estimates by the following [[recursion|recursive]] scheme 
{{NumBlk|:|<math> y_{n+1} = y_n + hf(t_n,y_n). </math>|{{EquationRef|4}}}}
This is the ''[[Euler method]]'' (or ''[[forward Euler method]]'', in contrast with the ''backward Euler method'', to be described below). The method is named after [[Leonhard Euler]] who described it in 1768.

The Euler method is an example of an [[explicit and implicit methods|''explicit'']] method. This means that the new value ''y''<sub>''n''+1</sub> is defined in terms of things that are already known, like ''y''<sub>''n''</sub>.

===Backward Euler method===
{{details|Backward Euler method}}
If, instead of ({{EquationNote|2}}), we use the approximation
{{NumBlk|:|<math> y'(t) \approx \frac{y(t) - y(t-h)}{h}, </math>|{{EquationRef|5}}}}
we get the ''backward Euler method'':
{{NumBlk|:|<math> y_{n+1} = y_n + hf(t_{n+1},y_{n+1}). </math>|{{EquationRef|6}}}}
The backward Euler method is an [[explicit and implicit methods|''implicit'']] method, meaning that we have to solve an equation to find ''y''<sub>''n''+1</sub>. One often uses [[fixed-point iteration]] or (some modification of) the [[Newton's method|Newton–Raphson method]] to achieve this.

It costs more time to solve this equation than explicit methods; this cost must be taken into consideration when one selects the method to use. The advantage of implicit methods such as ({{EquationNote|6}}) is that they are usually more stable for solving a [[stiff equation]], meaning that a larger step size ''h'' can be used.

===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
{{NumBlk|:|<math>y'(t) = -A\, y+ \mathcal{N}(y), </math>|{{EquationRef|7}}}}
or it has been locally linearized about a background state to produce a linear term <math>-Ay</math> and a nonlinear term <math>\mathcal{N}(y)</math>.

Exponential integrators are constructed by multiplying ({{EquationNote|7}}) by <math display="inline">e^{A t}</math>, and exactly integrating the result over
a time interval <math>[t_n, t_{n+1} = t_n + h]</math>:
:<math> y_{n+1} = e^{-A h } y_n + \int_{0}^{h} e^{ -(h-\tau) A } \mathcal{N}\left( y\left( t_n+\tau \right) \right)\, d\tau. </math>
This integral equation is exact, but it doesn't define the integral.

The first-order exponential integrator can be realized by holding <math>\mathcal{N}( y( t_n+\tau ) )</math> constant over the full interval:
{{NumBlk|:|<math>y_{n+1} = e^{-Ah}y_n + A^{-1}(1-e^{-Ah}) \mathcal{N}( y( t_n ) )\ .</math>|{{EquationRef|8}}}}

===Generalizations===
The Euler method is often not accurate enough. In more precise terms, it only has order one (the concept of ''order'' is explained below). This caused mathematicians to look for higher-order methods.

One possibility is to use not only the previously computed value ''y''<sub>''n''</sub> to determine ''y''<sub>''n''+1</sub>, but to make the solution depend on more past values. This yields a so-called ''multistep method''. Perhaps the simplest is the [[leapfrog method]] which is second order and (roughly speaking) relies on two time values.

Almost all practical multistep methods fall within the family of [[linear multistep method]]s, which have the form
<math display="block">\begin{align}
&{} \alpha_k y_{n+k} + \alpha_{k-1} y_{n+k-1} + \cdots + \alpha_0 y_n \\
&{} \quad = h \left[ \beta_k f(t_{n+k},y_{n+k}) + \beta_{k-1} f(t_{n+k-1},y_{n+k-1}) + \cdots + \beta_0 f(t_n,y_n) \right].
\end{align}</math>

Another possibility is to use more points in the interval <math>[t_n,t_{n+1}]</math>. This leads to the family of [[Runge–Kutta methods]], named after [[Carl David Tolmé Runge|Carl Runge]] and [[Martin Kutta]]. One of their fourth-order methods is especially popular.

===Advanced features===
A good implementation of one of these methods for solving an ODE entails more than the time-stepping formula.

It is often inefficient to use the same step size all the time, so ''variable step-size methods'' have been developed. Usually, the step size is chosen such that the (local) error per step is below some tolerance level. This means that the methods must also compute an ''error indicator'', an estimate of the local error.

An extension of this idea is to choose dynamically between different methods of different orders (this is called a ''variable order method''). Methods based on [[Richardson extrapolation]],<ref>Brezinski, C., & Zaglia, M. R. (2013). Extrapolation methods: theory and practice. Elsevier.</ref> such as the [[Bulirsch–Stoer algorithm]],<ref>Monroe, J. L. (2002). Extrapolation and the Bulirsch-Stoer algorithm. Physical Review E, 65(6), 066116.</ref><ref>Kirpekar, S. (2003). Implementation of the Bulirsch Stoer extrapolation method. Department of Mechanical Engineering, UC Berkeley/California.</ref> are often used to construct various methods of different orders.

Other desirable features include: 
* ''dense output'': cheap numerical approximations for the whole integration interval, and not only at the points ''t''<sub>0</sub>, ''t''<sub>1</sub>, ''t''<sub>2</sub>, ...
* ''event location'': finding the times where, say, a particular function vanishes. This typically requires the use of a [[root-finding algorithm]].
* support for [[parallel computing]].
* when used for integrating with respect to time, time reversibility

===Alternative methods===
Many methods do not fall within the framework discussed here. Some classes of alternative methods are:
* ''multiderivative methods'', which use not only the function ''f'' but also its derivatives. This class includes ''Hermite–Obreschkoff methods'' and ''[[Runge–Kutta–Fehlberg method|Fehlberg methods]]'', as well as methods like the [[Parker–Sochacki method]]<ref>Nurminskii, E. A., & Buryi, A. A. (2011). Parker-Sochacki method for solving systems of ordinary differential equations using graphics processors. Numerical Analysis and Applications, 4(3), 223.</ref> or Bychkov–Scherbakov method, which compute the coefficients of the [[Taylor series]] of the solution ''y'' recursively.
* ''methods for second order ODEs''. We said that all higher-order ODEs can be transformed to first-order ODEs of the form&nbsp;(1). While this is certainly true, it may not be the best way to proceed. In particular, ''[[Nyström method]]s'' work directly with second-order equations.
* ''[[geometric integrator|geometric integration methods]]''<ref>Hairer, E., Lubich, C., & Wanner, G. (2006). Geometric numerical integration: structure-preserving algorithms for ordinary differential equations (Vol. 31). Springer Science & Business Media.</ref><ref>Hairer, E., Lubich, C., & Wanner, G. (2003). Geometric numerical integration illustrated by the Störmer–Verlet method. Acta Numerica, 12, 399-450.</ref> are especially designed for special classes of ODEs (for example, [[symplectic integrator]]s for the solution of [[Hamiltonian mechanics|Hamiltonian equations]]). They take care that the numerical solution respects the underlying structure or geometry of these classes.
* ''[[Quantized state systems methods]]'' are a family of ODE integration methods based on the idea of state quantization. They are efficient when simulating sparse systems with frequent discontinuities.

===Parallel-in-time methods===
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> 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 |access-date=15 November 2023 |website=Parallel-in-Time.org}}</ref>

== Analysis ==

[[Numerical analysis]] is not only the design of numerical methods, but also their analysis. Three central concepts in this analysis are:
* ''convergence'': whether the method approximates the solution,
* ''order'': how well it approximates the solution, and
* [[numerical stability|''stability'']]: whether errors are damped out.<ref>Higham, N. J. (2002). Accuracy and stability of numerical algorithms (Vol. 80). SIAM.</ref>

===Convergence===
{{main|Sequence|Limit (mathematics)|Limit of a sequence}}
A numerical method is said to be ''convergent'' if the numerical solution approaches the exact solution as the step size ''h'' goes to 0. More precisely, we require that for every ODE (1) with a [[Lipschitz continuous|Lipschitz]] function ''f'' and every ''t''<sup>*</sup>&nbsp;>&nbsp;0,

:<math> \lim_{h\to0^+} \max_{n=0,1,\dots,\lfloor t^*/h\rfloor} \left\| y_{n,h} - y(t_n) \right\| = 0. </math>

All the methods mentioned above are convergent.

===Consistency and order===
{{details|Truncation error (numerical integration)}}
Suppose the numerical method is

:<math> y_{n+k} = \Psi(t_{n+k}; y_n, y_{n+1}, \dots, y_{n+k-1}; h). \, </math>

The ''local (truncation) error'' of the method is the error committed by one step of the method. That is, it is the difference between the result given by the method, assuming that no error was made in earlier steps, and the exact solution:

:<math> \delta^h_{n+k} = \Psi \left( t_{n+k}; y(t_n), y(t_{n+1}), \dots, y(t_{n+k-1}); h \right) - y(t_{n+k}). </math>

The method is said to be ''consistent'' if 
:<math> \lim_{h\to 0} \frac{\delta^h_{n+k}}{h} = 0. </math>
The method has ''order'' <math>p</math> if
:<math> \delta^h_{n+k} = O(h^{p+1}) \quad\mbox{as } h\to0. </math>
Hence a method is consistent if it has an order greater than 0. The (forward) Euler method (4) and the backward Euler method (6) introduced above both have order 1, so they are consistent. Most methods being used in practice attain higher order. Consistency is a necessary condition for convergence{{Citation needed|date=September 2019}}, but not sufficient; for a method to be convergent, it must be both consistent and [[zero-stable]].

A related concept is the ''global (truncation) error'', the error sustained in all the steps one needs to reach a fixed time <math>t</math>. Explicitly, the global error at time <math>t</math>'' is <math>y_N - y(t)</math> where <math>N = (t - t_0)/h</math>. The global error of a <math>p</math>''th order one-step method is <math>O(h^p)</math>; in particular, such a method is convergent.  This statement is not necessarily true for multi-step methods.

===Stability and stiffness===
{{details|Stiff equation}}
For some differential equations, application of standard methods—such as the Euler method, explicit [[Runge–Kutta methods]], or [[multistep method]]s (for example, Adams–Bashforth methods)—exhibit instability in the solutions, though other methods may produce stable solutions. This "difficult behaviour" in the equation (which may not necessarily be complex itself) is described as ''stiffness'', and is often caused by the presence of different time scales in the underlying problem.<ref>Miranker, A. (2001). Numerical Methods for Stiff Equations and Singular Perturbation Problems: and singular perturbation problems (Vol. 5). Springer Science & Business Media.</ref> For example, a collision in a mechanical system like in an [[impact oscillator]] typically occurs at much smaller time scale than the time for the motion of objects; this discrepancy makes for very "sharp turns" in the curves of the state parameters.

Stiff problems are ubiquitous in [[chemical kinetics]], [[control theory]], [[solid mechanics]], [[weather forecasting]], [[biology]], [[plasma physics]], and [[electronics]]. One way to overcome stiffness is to extend the notion of differential equation to that of [[differential inclusion]], which allows for and models non-smoothness.<ref name="Fiedler2001">{{cite book|editor=Bernold Fiedler|title=Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems|year=2001|publisher=Springer Science & Business Media|isbn=978-3-540-41290-8|page=431|chapter=Non-smooth Dynamical Systems: An Overview|author1=Markus Kunze |author2=Tassilo Kupper }}</ref><ref name="ZanderSchieferdecker2011">{{cite book|editor=Justyna Zander, Ina Schieferdecker and Pieter J. Mosterman|title=Model-Based Testing for Embedded Systems|year=2011|publisher=CRC Press|isbn=978-1-4398-1845-9|page=411|author=Thao Dang|chapter=Model-Based Testing of Hybrid Systems}}</ref>

== History ==

Below is a [[Chronology|timeline]] of some important developments in this field.<ref>Brezinski, C., & Wuytack, L. (2012). Numerical analysis: Historical developments in the 20th century. Elsevier.</ref><ref>Butcher, J. C. (1996). A history of Runge-Kutta methods. Applied numerical mathematics, 20(3), 247-260.</ref>

* 1768 - [[Leonhard Euler]] publishes his method.
* 1824 - [[Augustin Louis Cauchy]] proves convergence of the Euler method. In this proof, Cauchy uses the implicit Euler method.
* 1855 - First mention of the [[multistep method]]s of [[John Couch Adams]] in a letter written by [[Francis Bashforth]].
* 1895 - [[Carl David Tolmé Runge|Carl Runge]] publishes the first [[Runge–Kutta method]].
* 1901 - [[Martin Kutta]] describes the popular fourth-order [[Runge–Kutta method]].
* 1910 - [[Lewis Fry Richardson]] announces his [[extrapolation method]], [[Richardson extrapolation]].
* 1952 - [[Charles F. Curtiss]] and [[Joseph Oakland Hirschfelder]] coin the term ''[[stiff equation]]s''.
* 1963 - [[Germund Dahlquist]] introduces ''[[Stiff equation#A-stability|A-stability]]'' of integration methods.

== Numerical solutions to second-order one-dimensional boundary value problems ==

Boundary value problems (BVPs) are usually solved numerically by solving an approximately equivalent matrix problem obtained by discretizing the original BVP.<ref>Ascher, U. M., Mattheij, R. M., & Russell, R. D. (1995). Numerical solution of boundary value problems for ordinary differential equations. Society for Industrial and Applied Mathematics.</ref> The most commonly used method for numerically solving BVPs in one dimension is called the [[Finite Difference Method]].<ref name="fdm"/>  This method takes advantage of linear combinations of point values to construct [[finite difference coefficient]]s that describe derivatives of the function. For example, the second-order [[central difference]] approximation to the first derivative is given by:

: <math> \frac{u_{i+1}-u_{i-1}}{2h} = u'(x_i) + \mathcal{O}(h^2), </math>

and the second-order [[central difference]] for the second derivative is given by:

: <math> \frac{u_{i+1}- 2 u_i + u_{i-1}}{h^2} = u''(x_i) + \mathcal{O}(h^2). </math>

In both of these formulae, <math> h=x_i-x_{i-1}</math> is the distance between neighbouring ''x'' values on the discretized domain. One then constructs a linear system that can then be solved by standard [[numerical linear algebra|matrix methods]]. For example, suppose the equation to be solved is:

: <math>\begin{align}
&{} \frac{d^2 u}{dx^2} -u =0,\\
&{} u(0)=0, \\
&{} u(1)=1.
\end{align}</math>

The next step would be to discretize the problem and use linear derivative approximations such as

: <math> u''_i =\frac{u_{i+1}-2u_{i}+u_{i-1}}{h^2} </math>

and solve the resulting system of linear equations. This would lead to equations such as:

: <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]] that has non-trivial solutions.

== See also ==
* [[Courant–Friedrichs–Lewy condition]]
* [[Energy drift]]
* [[General linear methods]]
* [[List of numerical analysis topics#Numerical methods for ordinary differential equations]]
* [[Reversible reference system propagation algorithm]]
*[[Modelica]] Language and [[OpenModelica]] software

== Notes ==
{{reflist|2}}

== References ==
*{{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|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 (1671–1914)]'', 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. 
| title=An accurate numerical method and algorithm for constructing solutions of chaotic systems 
| journal=Journal of Applied Nonlinear Dynamics 
| year = 2020 | volume = 9 | issue = 2 | pages=207–221
| doi = 10.5890/JAND.2020.06.004 | arxiv=2011.10664 
| s2cid=225853788 
}}
*{{GitHub|mskashi/kv}} ([[C++]] library with rigorous ODE solvers)
*[http://www.ti3.tu-harburg.de/intlab/ INTLAB] (A library made by [[MATLAB]]/[[GNU Octave]] which includes rigorous ODE solvers)
{{Numerical integrators}}
{{Industrial and applied mathematics}}
{{DEFAULTSORT:Numerical Ordinary Differential Equations}}
[[Category:Numerical differential equations| ]]
[[Category:Ordinary differential equations]]