Illustration of numerical integration for the differential equation
y'=y, y(0)=1. Blue: the
Euler method, green: the
midpoint method, red: the exact solution,
y=e^t. The step size is
h=1.0.
The same illustration for h=0.25. It is seen that the midpoint method converges faster than the Euler method.
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term is sometimes taken to mean the computation of integrals.
Many differential equations cannot be solved using symbolic computation ("analysis"). For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms 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, for instance in physics, chemistry, biology, and economics. 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 firstorder differential equation is an Initial value problem (IVP) of the form,^{[1]}

y'(t) = f(t,y(t)), \qquad y(t_0)=y_0, \qquad\qquad (1)
where f is a function that maps [t_{0},∞) × R^{d} to R^{d}, and the initial condition y_{0} ∈ R^{d} is a given vector. Firstorder means that only the first derivative of y appears in the equation, and higher derivatives are absent.
Without loss of generality to higherorder systems, we restrict ourselves to firstorder differential equations, because a higherorder ODE can be converted into a larger system of firstorder equations by introducing extra variables. For example, the secondorder equation y'' = −y can be rewritten as two firstorder equations: y' = z and z' = −y.
In this section, we describe numerical methods for IVPs, and remark that boundary value problems (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 differences, Galerkin methods, or collocation methods are appropriate for that class of problems.
The Picard–Lindelöf theorem states that there is a unique solution, provided f is Lipschitzcontinuous.
Methods
Numerical methods for solving firstorder IVPs often fall into one of two large categories: linear multistep methods, or RungeKutta 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 methods include AdamsMoulton methods, and backward differentiation methods (BDF), whereas implicit RungeKutta methods^{[2]} include diagonally implicit RungeKutta (DIRK), singly diagonally implicit runge kutta (SDIRK), and GaussRadau (based on Gaussian quadrature) numerical methods. Explicit examples from the linear multistep family include the AdamsBashforth methods, and any RungeKutta method with a lower diagonal Butcher tableau is explicit. A loose rule of thumb dictates that stiff differential equations require the use of implicit schemes, whereas nonstiff problems can be solved more efficiently with explicit schemes.
The socalled general linear methods (GLMs) are a generalization of the above two large classes of methods.
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 (1), we replace the derivative y' by the finite difference approximation

y'(t) \approx \frac{y(t+h)  y(t)}{h}, \qquad\qquad (2)
which when rearranged yields the following formula

y(t+h) \approx y(t) + hy'(t) \qquad\qquad
and using (1) gives:

y(t+h) \approx y(t) + hf(t,y(t)). \qquad\qquad (3)
This formula is usually applied in the following way. We choose a step size h, and we construct the sequence t_{0}, t_{1} = t_{0} + h, t_{2} = t_{0} + 2h, … We denote by y_{n} a numerical estimate of the exact solution y(t_{n}). Motivated by (3), we compute these estimates by the following recursive scheme

y_{n+1} = y_n + hf(t_n,y_n). \qquad\qquad (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 method. This means that the new value y_{n+1} is defined in terms of things that are already known, like y_{n}.
Backward Euler method
If, instead of (2), we use the approximation

y'(t) \approx \frac{y(t)  y(th)}{h}, \qquad\qquad (5)
we get the backward Euler method:

y_{n+1} = y_n + hf(t_{n+1},y_{n+1}). \qquad\qquad (6)
The backward Euler method is an implicit method, meaning that we have to solve an equation to find y_{n+1}. One often uses fixed point iteration or (some modification of) the 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 (6) is that they are usually more stable for solving a stiff equation, meaning that a larger step size h can be used.
Firstorder exponential integrator method
Exponential integrators describe a large class of integrators that have recently seen a lot of development.^{[3]} They date back to at least the 1960s.
In place of (1), we assume the differential equation is either of the form

y'(t) = A\, y+ \mathcal{N}(y), \qquad\qquad\qquad (7)
or it has been locally linearize about a background state to produce a linear term Ay and a nonlinear term \mathcal{N}(y).
Exponential integrators are constructed by multiplying (7) by e^{A t}, and exactly integrating the result over a time interval [t_n, t_{n+1} = t_n + h]:

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.
This approximation is exact, but it doesn't define the integral.
The firstorder exponential integrator can be realized by holding \mathcal{N}( y( t_n+\tau ) ) constant over the full interval:

y_{n+1} = e^{Ah}y_n + A^{1}(1e^{Ah}) \mathcal{N}( y( t_n ) )\ . \qquad\qquad (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 higherorder methods.
One possibility is to use not only the previously computed value y_{n} to determine y_{n+1}, but to make the solution depend on more past values. This yields a socalled 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 methods, which have the form

\alpha_k y_{n+k} + \alpha_{k1} y_{n+k1} + \cdots + \alpha_0 y_n

= h \left[ \beta_k f(t_{n+k},y_{n+k}) + \beta_{k1} f(t_{n+k1},y_{n+k1}) + \cdots + \beta_0 f(t_n,y_n) \right].
Another possibility is to use more points in the interval [t_{n},t_{n+1}]. This leads to the family of Runge–Kutta methods, named after Carl Runge and Martin Kutta. One of their fourthorder methods is especially popular.
Advanced features
A good implementation of one of these methods for solving an ODE entails more than the timestepping formula.
It is often inefficient to use the same step size all the time, so variable stepsize 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, such as the Bulirsch–Stoer algorithm, 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_{0}, t_{1}, t_{2}, ...

event location: finding the times where, say, a particular function vanishes. This typically requires the use of a rootfinding 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 Fehlberg methods, as well as methods like the Parker–Sochacki method or BychkovScherbakov method, which compute the coefficients of the Taylor series of the solution y recursively.

methods for second order ODEs. We said that all higherorder ODEs can be transformed to firstorder ODEs of the form (1). While this is certainly true, it may not be the best way to proceed. In particular, Nyström methods work directly with secondorder equations.

geometric integration methods are especially designed for special classes of ODEs (e.g., symplectic integrators for the solution of 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.
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

stability: whether errors are damped out.
Convergence
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 function f and every t^{*} > 0,

\lim_{h\to0+} \max_{n=0,1,\dots,\lfloor t^*/h\rfloor} \ y_{n,h}  y(t_n) \ = 0.
All the methods mentioned above are convergent. In fact, a numerical scheme has to be convergent to be of any use.
Consistency and order
Suppose the numerical method is

y_{n+k} = \Psi(t_{n+k}; y_n, y_{n+1}, \dots, y_{n+k1}; h). \,
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:

\delta^h_{n+k} = \Psi \left( t_{n+k}; y(t_n), y(t_{n+1}), \dots, y(t_{n+k1}); h \right)  y(t_{n+k}).
The method is said to be consistent if

\lim_{h\to 0} \frac{\delta^h_{n+k}}{h} = 0.
The method has order p if

\delta^h_{n+k} = O(h^{p+1}) \quad\mbox{as } h\to0.
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, but not sufficient; for a method to be convergent, it must be both consistent and zerostable.
A related concept is the global (truncation) error, the error sustained in all the steps one needs to reach a fixed time t. Explicitly, the global error at time t is y_{N} − y(t) where N = (t−t_{0})/h. The global error of a pth order onestep method is O(h^{p}); in particular, such a method is convergent. This statement is not necessarily true for multistep methods.
Stability and stiffness
For some differential equations, application of standard methods —such as the Euler method, explicit Runge–Kutta methods, or multistep methods (e.g., 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. Stiff problems are ubiquitous in chemical kinetics, control theory, solid mechanics, weather forecasting, biology, plasma physics, and electronics.
History
Below is a timeline of some important developments in this field.
Numerical solutions to secondorder onedimensional boundary value problems
Boundary value problems (BVPs) are usually solved numerically by solving an approximately equivalent matrix problem obtained by discretizing the original BVP. The most commonly used method for numerically solving BVPs in one dimension is called the Finite Difference Method. This method takes advantage of linear combinations of point values to construct finite difference coefficients that describe derivatives of the function. For example, the secondorder central difference approximation to the first derivative is given by:

\frac{u_{i+1}u_{i1}}{2h} = u'(x_i) + \mathcal{O}(h^2),
and the secondorder central difference for the second derivative is given by:

\frac{u_{i+1} 2 u_i + u_{i1}}{h^2} = u''(x_i) + \mathcal{O}(h^2).
In both of these formulae, h=x_ix_{i1} is the distance between neighbouring x values on the discretized domain. One then constructs a linear system that can then be solved by standard matrix methods. For instance, suppose the equation to be solved is:

\frac{d^2 u}{dx^2} u =0,

u(0)=0,

u(1)=1.
The next step would be to discretize the problem and use linear derivative approximations such as

u''_i =\frac{u_{i+1}2u_{i}+u_{i1}}{h^2}
and solve the resulting system of linear equations. This would lead to equations such as:

\frac{u_{i+1}2u_{i}+u_{i1}}{h^2}u_i = 0, \quad \forall i={1,2,3,...,n1}.
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 u(0)=u_0 and u(1)=u_n and since these two values are known, one can simply substitute them into this equation and as a result have a nonhomogeneous linear system of equations that has nontrivial solutions.
See also
Notes
References

Bradie, Brian (2006). A Friendly Introduction to Numerical Analysis. Upper Saddle River, New Jersey: Pearson Prentice Hall.

J. C. Butcher, Numerical methods for ordinary differential equations, ISBN 0471967580

Ernst Hairer, Syvert Paul Nørsett and Gerhard Wanner, Solving ordinary differential equations I: Nonstiff problems, second edition, Springer Verlag, Berlin, 1993. ISBN 3540566708.

Ernst Hairer and Gerhard Wanner, Solving ordinary differential equations II: Stiff and differentialalgebraic problems, second edition, Springer Verlag, Berlin, 1996. ISBN 3540604529.
(This twovolume monograph systematically covers all aspects of the field.)

Hochbruck, Marlis; Ostermann, Alexander (May 2010). "Exponential integrators". pp. 209–286.

Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 1996. ISBN 0521553768 (hardback), ISBN 0521556554 (paperback).
(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 0471929905.
(Textbook, slightly more demanding than the book by Iserles.)
External links

Joseph W. Rudmin, Application of the Parker–Sochacki Method to Celestial Mechanics, 1998.

Dominique Tournès, L'intégration approchée des équations différentielles ordinaires (16711914), 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 Englishlanguage material on the history of ODE numerical analysis, see e.g. the paper books by Chabert and Goldstine quoted by him.)
Numerical integration methods by order




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