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[[Image:Numerical integration illustration, h=1.png|right|thumb|Illustration of numerical integration for the differential equation <math>y'=y, y(0)=1.</math> Blue: the [[Euler method]], green: the [[midpoint method]], red: the exact solution, <math>y=e^t.</math> The step size is <math>h=1.0.</math>]] | |||
[[Image:Numerical integration illustration, h=0.25.png|right|thumb|The same illustration for <math>h=0.25.</math> It is seen that the midpoint method converges faster than the Euler method.]] | |||
This subject is the part of [[numerical analysis]] which studies the methods for finding numerical approximations to the solutions of [[ordinary differential equation]]s (ODEs). This field is also known under the name ''[[numerical integration]],'' but some people reserve this term for the computation of [[integral]]s. | |||
Many differential equations cannot be solved analytically; however, in science and engineering, a numeric approximation to the solution is often good enough to solve a problem. 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, 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 first-order differential equation is an [[Initial value problem]] (IVP) of the form,<ref>{{harvtxt|Bradie|2006|pp=533–655}}</ref> | |||
:<math>y'(t) = f(t,y(t)), \qquad y(t_0)=y_0, \qquad\qquad (1)</math> | |||
where ''f'' is a function that maps [''t''<sub>0</sub>,∞) × '''R'''<sup>d</sup> to '''R'''<sup>d</sup>, and the initial condition ''y''<sub>0</sub> ∈ '''R'''<sup>d</sup> 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 | |||
''y''<nowiki>''</nowiki> = −''y'' | |||
can be rewritten as two first-order equations: ''y''<nowiki>'</nowiki> = ''z'' and ''z''<nowiki>'</nowiki> = −''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, [[Galerkin method]]s, or [[collocation method]]s are appropriate for that class of problems. | |||
The [[Picard–Lindelöf theorem]] states that there is a unique solution, provided ''f'' is [[Lipschitz continuous]]. | |||
== Methods == | |||
Numerical methods for solving first-order IVPs often fall into one of two large categories: [[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 [[Runge-Kutta methods#Implicit Runge–Kutta methods|implicit Runge-Kutta methods]]<ref>{{harvtxt|Hairer|Nørsett|Wanner|1993|pages=204–215}}</ref> include [[diagonally implicit Runge-Kutta]] (DIRK), [[singly diagonally implicit runge kutta]] (SDIRK), and [[Gauss-Radau]] (based on [[Gaussian quadrature]]) numerical methods. | |||
Explicit examples from the [[Linear multistep method|linear multistep family]] include the | |||
[[Linear multistep method#Adams–Bashforth methods|Adams-Bashforth methods]], and any Runge-Kutta method | |||
with a lower diagonal [[Butcher tableau]] is [[List_of_Runge–Kutta_methods#Explicit_methods|explicit]]. | |||
A loose rule of thumb dictates that ''stiff'' differential equations require the use of implicit schemes, whereas non-stiff problems can be solved more efficiently with explicit schemes. | |||
We point out that the so-called [[general linear methods]] (GLMs) are a generalization of the above two large classes of methods. | |||
=== The Euler method ===<!-- This section is linked from [[Generating trigonometric tables]] --> | |||
{{details|Euler method}} | |||
A brief explanation: | |||
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. | |||
Rigorous development: | |||
Starting with the differential equation (1), we replace the derivative ''y''<nowiki>'</nowiki> by the [[finite difference]] approximation | |||
:<math> y'(t) \approx \frac{y(t+h) - y(t)}{h}, \qquad\qquad (2) </math> | |||
which when re-arranged yields the following formula | |||
:<math> y(t+h) \approx y(t) + hy'(t) \qquad\qquad </math> | |||
and using (1) gives: | |||
:<math> y(t+h) \approx y(t) + hf(t,y(t)). \qquad\qquad (3) </math> | |||
This formula is usually applied in the following way. We choose a step size ''h'', and we construct the sequence ''t''<sub>0</sub>, ''t''<sub>1</sub> = ''t''<sub>0</sub> + ''h'', ''t''<sub>2</sub> = ''t''<sub>0</sub> + 2''h'', … We denote by ''y''<sub>''n''</sub> a numerical estimate of the exact solution ''y''(''t''<sub>''n''</sub>). Motivated by (3), we compute these estimates by the following [[recursion|recursive]] scheme | |||
:<math> y_{n+1} = y_n + hf(t_n,y_n). \qquad\qquad (4) </math> | |||
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>. | |||
=== The backward Euler method === | |||
{{details|Backward Euler method}} | |||
If, instead of (2), we use the approximation | |||
:<math> y'(t) \approx \frac{y(t) - y(t-h)}{h}, \qquad\qquad (5)</math> | |||
we get the ''backward Euler method'': | |||
:<math> y_{n+1} = y_n + hf(t_{n+1},y_{n+1}). \qquad\qquad (6)</math> | |||
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. Of course, it costs time to solve this equation; 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.''' | |||
''' | |||
=== The first-order exponential integrator method === | |||
{{details|Exponential integrators}} | |||
Exponential integrators describe a large class of integrators that have recently seen a lot of development.<ref name="Exponential integrators">{{harvtxt|Hochbruck|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 (1), we assume the differential equation is either of the form | |||
:<math>y'(t) = -A\, y+ \mathcal{N}(y), \qquad\qquad\qquad (7)</math> | |||
or it has been locally linearize 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 (7) by <math>e^{A t}</math>, and exactly integrating the result over | |||
a time interval <math>[t^n, t^{n+1}]</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 approximation 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: | |||
:<math>y_{n+1} = e^{-Ah}y_n + A^{-1}(1-e^{-Ah}) \mathcal{N}( y( t_n ) )\ . \qquad\qquad (8)</math> | |||
=== 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> \alpha_k y_{n+k} + \alpha_{k-1} y_{n+k-1} + \cdots | |||
+ \alpha_0 y_n</math> | |||
::<math> = 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]. </math> | |||
Another possibility is to use more points in the interval [''t''<sub>''n''</sub>,''t''<sub>''n''+1</sub>]. This leads to the family of [[Runge–Kutta method]]s, named after [[Carl David Tolmé Runge|Carl Runge]] and [[Martin Wilhelm Kutta|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]], 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''<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]] 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 (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]]'' are especially designed for special classes of ODEs (e.g., [[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 simulation 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 | |||
* [[numerical stability|''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 continuous|Lipschitz]] function ''f'' and every ''t''<sup>*</sup> > 0, | |||
:<math> \lim_{h\to0+} \max_{n=0,1,\dots,\lfloor t^*/h\rfloor} \| y_{n,h} - y(t_n) \| = 0. </math> | |||
All the methods mentioned above are convergent. In fact, a numerical scheme has to be convergent to be of any use. | |||
=== 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, 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 ''t''. Explicitly, the global error at time ''t'' is ''y''<sub>''N''</sub> − ''y''(''t'') where ''N'' = (''t''−''t''<sub>0</sub>)/''h''. The global error of a ''p''th order one-step method is O(''h''<sup>''p''</sup>); 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 (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 [[Chronology|timeline]] of some important developments in this field. | |||
* 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 [[F. Bashforth]]. | |||
* 1895 - [[Carl David Tolmé Runge|Carl Runge]] publishes the first [[Runge–Kutta method]]. | |||
* 1905 - [[Martin Wilhelm Kutta|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. 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 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 instance, suppose the equation to be solved is: | |||
: <math> \frac{d^2 u}{dx^2} -u =0, </math> | |||
: <math> u(0)=0, </math> | |||
: <math> u(1)=1. </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-homogenous linear system of 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]] | |||
== Notes == | |||
<references/> | |||
== References == | |||
{{Reflist}} | |||
*{{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=0-13-013054-0}} | |||
*[[John C. Butcher|J. C. Butcher]], ''Numerical methods for ordinary differential equations'', ISBN 0-471-96758-0 | |||
*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. | |||
*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|coauthors=Ostermann, Alexander|title=Exponential integrators|date=May 2010|year=2010|pages=209–286|doi=10.1017/S0962492910000048|url=http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=7701740}} | |||
*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]'', 1998. | |||
* Dominique Tournès, ''[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 e.g. the paper books by Chabert and Goldstine quoted by him.) | |||
{{Numerical integrators}} | |||
{{DEFAULTSORT:Numerical Ordinary Differential Equations}} | |||
[[Category:Numerical differential equations| ]] | |||
[[Category:Ordinary differential equations]] | |||
Revision as of 01:12, 4 February 2014
This subject is the part of numerical analysis which studies the methods for finding numerical approximations to the solutions of ordinary differential equations (ODEs). This field is also known under the name numerical integration, but some people reserve this term for the computation of integrals.
Many differential equations cannot be solved analytically; however, in science and engineering, a numeric approximation to the solution is often good enough to solve a problem. 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 first-order differential equation is an Initial value problem (IVP) of the form,[1]
where f is a function that maps [t0,∞) × Rd to Rd, and the initial condition y0 ∈ Rd 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 y'' = −y can be rewritten as two first-order 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 Lipschitz continuous.
Methods
Numerical methods for solving first-order IVPs often fall into one of two large categories: linear multistep methods, 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 methods include Adams-Moulton methods, and backward differentiation methods (BDF), whereas implicit Runge-Kutta methods[2] include diagonally implicit Runge-Kutta (DIRK), singly diagonally implicit runge kutta (SDIRK), and Gauss-Radau (based on Gaussian quadrature) numerical methods. Explicit examples from the linear multistep family include the Adams-Bashforth methods, and any Runge-Kutta 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 non-stiff problems can be solved more efficiently with explicit schemes.
We point out that the so-called general linear methods (GLMs) are a generalization of the above two large classes of methods.
The Euler method
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i) registered as a patent agent or its equal in any nation or territory, or by a patent workplace, specified within the Fourth Schedule; The business-specific tips for the true property agency and telecommunication sectors have been crafted to address considerations about scenarios that particularly apply to the two sectors, the PDPC stated. Mr Steven Tan, Managing Director of OrangeTee real property company, nonetheless, felt that it was a matter of "practising until it becomes part of our knowledge". "After a while, the agents ought to know the spirit behind the (Act)," he stated. Rising office sector leads real property market efficiency, while prime retail and enterprise park segments moderate and residential sector continues in decline Please choose an attendee for donation.
A brief explanation: 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.
Rigorous development: Starting with the differential equation (1), we replace the derivative y' by the finite difference approximation
which when re-arranged yields the following formula
and using (1) gives:
This formula is usually applied in the following way. We choose a step size h, and we construct the sequence t0, t1 = t0 + h, t2 = t0 + 2h, … We denote by yn a numerical estimate of the exact solution y(tn). Motivated by (3), we compute these estimates by the following recursive scheme
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 yn+1 is defined in terms of things that are already known, like yn.
The backward Euler method
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Step one in direction of conducting enterprise as an actual property company in Singapore is to include an organization, or if you happen to're going the partnership or sole-proprietorship route, register your Limited Legal responsibility Partnership or sole-proprietorship with the ACRA (Accounting and Company Regulatory Authority of Singapore) Whether or not you might be considering to promote, let, hire or buy a new industrial property, we're right here to assist. Search and browse our commercial property section. Possess not less than 3 years of working expertise below a Singapore licensed real-property agency; Sale, letting and property administration and taxation companies. three Shenton Means, #10-08 Shenton Home, Singapore 068805. Real property agents for purchasing, promoting, leasing, and renting property. Caveat Search
Firstly, the events might take into account to rescind the sale and buy agreement altogether. This avenue places the contracting events to a position as if the contract didn't happen. It's as if the contract was terminated from the start and events are put back into place that they were before the contract. Any items or monies handed are returned to the respective original house owners. As the worldwide real property market turns into extra refined and worldwide real property investments will increase, the ERA real estate network is well equipped to offer professional recommendation and guidance to our shoppers in making critical actual estate decisions. Relocationg, leasing and sales of properties for housing, food and beverage, retail and workplace wants.
Pasir Panjang, Singapore - $5,000-6,000 per 30 days By likelihood one among our buddies here in Singapore is an agent and we made contact for her to help us locate an residence, which she did. days from the date of execution if the doc is signed in Singapore; Be a Singapore Citizen or PR (Permanent Resident); The regulations also prohibit property agents from referring their shoppers to moneylenders, to discourage irresponsible shopping for. Brokers are additionally prohibited from holding or dealing with money on behalf of any party in relation to the sale or purchase of any property situated in Singapore, and the lease of HDB property. - Negotiate To Close A Sale together with sale and lease of HDB and private properties) Preparing your house for sale FEATURED COMMERCIAL AGENTS Property Guides
i) registered as a patent agent or its equal in any nation or territory, or by a patent workplace, specified within the Fourth Schedule; The business-specific tips for the true property agency and telecommunication sectors have been crafted to address considerations about scenarios that particularly apply to the two sectors, the PDPC stated. Mr Steven Tan, Managing Director of OrangeTee real property company, nonetheless, felt that it was a matter of "practising until it becomes part of our knowledge". "After a while, the agents ought to know the spirit behind the (Act)," he stated. Rising office sector leads real property market efficiency, while prime retail and enterprise park segments moderate and residential sector continues in decline Please choose an attendee for donation.
If, instead of (2), we use the approximation
we get the backward Euler method:
The backward Euler method is an implicit method, meaning that we have to solve an equation to find yn+1. One often uses fixed point iteration or (some modification of) the Newton–Raphson method to achieve this. Of course, it costs time to solve this equation; 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.
The first-order exponential integrator method
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Actual estate agency specialising in non-public condos and landed properties island vast. 10 Winstedt Highway, District 10, #01-thirteen, Singapore 227977. Property providers for enterprise relocation. Situated at 371 Beach Street, #19-10 KeyPoint, Singapore 199597. Property agents for homes, town houses, landed property, residences and condominium for sales and rentals of properties. Administration letting services for property homeowners. is there a single authority in singapore who regulates real property agents that i can file a complaint with for unethical behaviour? or is CASE is simply route? The 188 pages of Secrets and techniques of Singapore Property Gurus are full of professional knowledge and life altering wisdom. Asian industrial property market outlook Property Listing Supervisor Property Advertising Services
Should sellers go along with an agent who claims to specialize in your space? His experience might turn out to be useful, but he is probably additionally advertising a number of models within the neighbourhood – and so they're all your rivals. Within the worst-case state of affairs, your house may be used as a "showflat" as house owner YS Liang found. "Weekend after weekend, our agent would convey a stream of individuals to speed-go to our apartment, leaving within minutes. She did not even try to promote our condominium. It felt like we were just one of the many tour stops for her clients," he complains.
Step one in direction of conducting enterprise as an actual property company in Singapore is to include an organization, or if you happen to're going the partnership or sole-proprietorship route, register your Limited Legal responsibility Partnership or sole-proprietorship with the ACRA (Accounting and Company Regulatory Authority of Singapore) Whether or not you might be considering to promote, let, hire or buy a new industrial property, we're right here to assist. Search and browse our commercial property section. Possess not less than 3 years of working expertise below a Singapore licensed real-property agency; Sale, letting and property administration and taxation companies. three Shenton Means, #10-08 Shenton Home, Singapore 068805. Real property agents for purchasing, promoting, leasing, and renting property. Caveat Search
Firstly, the events might take into account to rescind the sale and buy agreement altogether. This avenue places the contracting events to a position as if the contract didn't happen. It's as if the contract was terminated from the start and events are put back into place that they were before the contract. Any items or monies handed are returned to the respective original house owners. As the worldwide real property market turns into extra refined and worldwide real property investments will increase, the ERA real estate network is well equipped to offer professional recommendation and guidance to our shoppers in making critical actual estate decisions. Relocationg, leasing and sales of properties for housing, food and beverage, retail and workplace wants.
Pasir Panjang, Singapore - $5,000-6,000 per 30 days By likelihood one among our buddies here in Singapore is an agent and we made contact for her to help us locate an residence, which she did. days from the date of execution if the doc is signed in Singapore; Be a Singapore Citizen or PR (Permanent Resident); The regulations also prohibit property agents from referring their shoppers to moneylenders, to discourage irresponsible shopping for. Brokers are additionally prohibited from holding or dealing with money on behalf of any party in relation to the sale or purchase of any property situated in Singapore, and the lease of HDB property. - Negotiate To Close A Sale together with sale and lease of HDB and private properties) Preparing your house for sale FEATURED COMMERCIAL AGENTS Property Guides
i) registered as a patent agent or its equal in any nation or territory, or by a patent workplace, specified within the Fourth Schedule; The business-specific tips for the true property agency and telecommunication sectors have been crafted to address considerations about scenarios that particularly apply to the two sectors, the PDPC stated. Mr Steven Tan, Managing Director of OrangeTee real property company, nonetheless, felt that it was a matter of "practising until it becomes part of our knowledge". "After a while, the agents ought to know the spirit behind the (Act)," he stated. Rising office sector leads real property market efficiency, while prime retail and enterprise park segments moderate and residential sector continues in decline Please choose an attendee for donation.
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
or it has been locally linearize about a background state to produce a linear term and a nonlinear term .
Exponential integrators are constructed by multiplying (7) by , and exactly integrating the result over a time interval :
![[t^{n},t^{{n+1}}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d672a6c069e35ec6b55a534e44ac1fe4877f97c)
This approximation is exact, but it doesn't define the integral.
The first-order exponential integrator can be realized by holding constant over the full interval:
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 yn to determine yn+1, 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 methods, which have the form
![{\displaystyle =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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b969bc5ad098113fc6bf87e1500c27cc26e6f31)
Another possibility is to use more points in the interval [tn,tn+1]. This leads to the family of Runge–Kutta methods, named after 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, 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 t0, t1, t2, ...
- 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 Fehlberg methods, as well as methods like the Parker–Sochacki method 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 (1). While this is certainly true, it may not be the best way to proceed. In particular, Nyström methods work directly with second-order 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 simulation 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,
All the methods mentioned above are convergent. In fact, a numerical scheme has to be convergent to be of any use.
Consistency and order
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Actual estate agency specialising in non-public condos and landed properties island vast. 10 Winstedt Highway, District 10, #01-thirteen, Singapore 227977. Property providers for enterprise relocation. Situated at 371 Beach Street, #19-10 KeyPoint, Singapore 199597. Property agents for homes, town houses, landed property, residences and condominium for sales and rentals of properties. Administration letting services for property homeowners. is there a single authority in singapore who regulates real property agents that i can file a complaint with for unethical behaviour? or is CASE is simply route? The 188 pages of Secrets and techniques of Singapore Property Gurus are full of professional knowledge and life altering wisdom. Asian industrial property market outlook Property Listing Supervisor Property Advertising Services
Should sellers go along with an agent who claims to specialize in your space? His experience might turn out to be useful, but he is probably additionally advertising a number of models within the neighbourhood – and so they're all your rivals. Within the worst-case state of affairs, your house may be used as a "showflat" as house owner YS Liang found. "Weekend after weekend, our agent would convey a stream of individuals to speed-go to our apartment, leaving within minutes. She did not even try to promote our condominium. It felt like we were just one of the many tour stops for her clients," he complains.
Step one in direction of conducting enterprise as an actual property company in Singapore is to include an organization, or if you happen to're going the partnership or sole-proprietorship route, register your Limited Legal responsibility Partnership or sole-proprietorship with the ACRA (Accounting and Company Regulatory Authority of Singapore) Whether or not you might be considering to promote, let, hire or buy a new industrial property, we're right here to assist. Search and browse our commercial property section. Possess not less than 3 years of working expertise below a Singapore licensed real-property agency; Sale, letting and property administration and taxation companies. three Shenton Means, #10-08 Shenton Home, Singapore 068805. Real property agents for purchasing, promoting, leasing, and renting property. Caveat Search
Firstly, the events might take into account to rescind the sale and buy agreement altogether. This avenue places the contracting events to a position as if the contract didn't happen. It's as if the contract was terminated from the start and events are put back into place that they were before the contract. Any items or monies handed are returned to the respective original house owners. As the worldwide real property market turns into extra refined and worldwide real property investments will increase, the ERA real estate network is well equipped to offer professional recommendation and guidance to our shoppers in making critical actual estate decisions. Relocationg, leasing and sales of properties for housing, food and beverage, retail and workplace wants.
Pasir Panjang, Singapore - $5,000-6,000 per 30 days By likelihood one among our buddies here in Singapore is an agent and we made contact for her to help us locate an residence, which she did. days from the date of execution if the doc is signed in Singapore; Be a Singapore Citizen or PR (Permanent Resident); The regulations also prohibit property agents from referring their shoppers to moneylenders, to discourage irresponsible shopping for. Brokers are additionally prohibited from holding or dealing with money on behalf of any party in relation to the sale or purchase of any property situated in Singapore, and the lease of HDB property. - Negotiate To Close A Sale together with sale and lease of HDB and private properties) Preparing your house for sale FEATURED COMMERCIAL AGENTS Property Guides
i) registered as a patent agent or its equal in any nation or territory, or by a patent workplace, specified within the Fourth Schedule; The business-specific tips for the true property agency and telecommunication sectors have been crafted to address considerations about scenarios that particularly apply to the two sectors, the PDPC stated. Mr Steven Tan, Managing Director of OrangeTee real property company, nonetheless, felt that it was a matter of "practising until it becomes part of our knowledge". "After a while, the agents ought to know the spirit behind the (Act)," he stated. Rising office sector leads real property market efficiency, while prime retail and enterprise park segments moderate and residential sector continues in decline Please choose an attendee for donation.
Suppose the numerical method is
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:
The method is said to be consistent if
The method has order if
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 zero-stable.
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 yN − y(t) where N = (t−t0)/h. The global error of a pth order one-step method is O(hp); in particular, such a method is convergent. This statement is not necessarily true for multi-step methods.
Stability and stiffness
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Actual estate agency specialising in non-public condos and landed properties island vast. 10 Winstedt Highway, District 10, #01-thirteen, Singapore 227977. Property providers for enterprise relocation. Situated at 371 Beach Street, #19-10 KeyPoint, Singapore 199597. Property agents for homes, town houses, landed property, residences and condominium for sales and rentals of properties. Administration letting services for property homeowners. is there a single authority in singapore who regulates real property agents that i can file a complaint with for unethical behaviour? or is CASE is simply route? The 188 pages of Secrets and techniques of Singapore Property Gurus are full of professional knowledge and life altering wisdom. Asian industrial property market outlook Property Listing Supervisor Property Advertising Services
Should sellers go along with an agent who claims to specialize in your space? His experience might turn out to be useful, but he is probably additionally advertising a number of models within the neighbourhood – and so they're all your rivals. Within the worst-case state of affairs, your house may be used as a "showflat" as house owner YS Liang found. "Weekend after weekend, our agent would convey a stream of individuals to speed-go to our apartment, leaving within minutes. She did not even try to promote our condominium. It felt like we were just one of the many tour stops for her clients," he complains.
Step one in direction of conducting enterprise as an actual property company in Singapore is to include an organization, or if you happen to're going the partnership or sole-proprietorship route, register your Limited Legal responsibility Partnership or sole-proprietorship with the ACRA (Accounting and Company Regulatory Authority of Singapore) Whether or not you might be considering to promote, let, hire or buy a new industrial property, we're right here to assist. Search and browse our commercial property section. Possess not less than 3 years of working expertise below a Singapore licensed real-property agency; Sale, letting and property administration and taxation companies. three Shenton Means, #10-08 Shenton Home, Singapore 068805. Real property agents for purchasing, promoting, leasing, and renting property. Caveat Search
Firstly, the events might take into account to rescind the sale and buy agreement altogether. This avenue places the contracting events to a position as if the contract didn't happen. It's as if the contract was terminated from the start and events are put back into place that they were before the contract. Any items or monies handed are returned to the respective original house owners. As the worldwide real property market turns into extra refined and worldwide real property investments will increase, the ERA real estate network is well equipped to offer professional recommendation and guidance to our shoppers in making critical actual estate decisions. Relocationg, leasing and sales of properties for housing, food and beverage, retail and workplace wants.
Pasir Panjang, Singapore - $5,000-6,000 per 30 days By likelihood one among our buddies here in Singapore is an agent and we made contact for her to help us locate an residence, which she did. days from the date of execution if the doc is signed in Singapore; Be a Singapore Citizen or PR (Permanent Resident); The regulations also prohibit property agents from referring their shoppers to moneylenders, to discourage irresponsible shopping for. Brokers are additionally prohibited from holding or dealing with money on behalf of any party in relation to the sale or purchase of any property situated in Singapore, and the lease of HDB property. - Negotiate To Close A Sale together with sale and lease of HDB and private properties) Preparing your house for sale FEATURED COMMERCIAL AGENTS Property Guides
i) registered as a patent agent or its equal in any nation or territory, or by a patent workplace, specified within the Fourth Schedule; The business-specific tips for the true property agency and telecommunication sectors have been crafted to address considerations about scenarios that particularly apply to the two sectors, the PDPC stated. Mr Steven Tan, Managing Director of OrangeTee real property company, nonetheless, felt that it was a matter of "practising until it becomes part of our knowledge". "After a while, the agents ought to know the spirit behind the (Act)," he stated. Rising office sector leads real property market efficiency, while prime retail and enterprise park segments moderate and residential sector continues in decline Please choose an attendee for donation.
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.
- 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 methods of John Couch Adams in a letter written by F. Bashforth.
- 1895 - Carl Runge publishes the first Runge–Kutta method.
- 1905 - 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 equations.
- 1963 - Germund Dahlquist introduces 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. 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 second-order central difference approximation to the first derivative is given by:
and the second-order central difference for the second derivative is given by:
In both of these formulae, 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:
The next step would be to discretize the problem and use linear derivative approximations such as
and solve the resulting system of linear equations. This would lead to equations such as:
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 and and since these two values are known, one can simply substitute them into this equation and as a result have a non-homogenous linear system of 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
Notes
- ↑ Template:Harvtxt
- ↑ Template:Harvtxt
- ↑ Template:Harvtxt This is a modern and extensive review paper for exponential integrators
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - J. C. Butcher, Numerical methods for ordinary differential equations, ISBN 0-471-96758-0
- 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.
- 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.
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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - 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).
(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.
(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 (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 e.g. the paper books by Chabert and Goldstine quoted by him.)