Log in / create accountArticle Talk Read Edit Search Numerical methods for ordinary differential equations From Wikipedia, the free encyclopedia (Redirected from Numerical ordinary differential equations) Main page Contents Featured content Current events Random article Donate to Wikipedia Interaction Help About Wikipedia Community portal Recent changes Contact Wikipedia Toolbox Print/export Languages Česky ﻓﺎرﺳﯽ Français Italiano This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations. (April 2010) Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution 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. Contents [hide] 1 The problem 2 Methods 2.1 The Euler method 2.2 The backward Euler method 2.3 The exponential Euler method 2.4 Generalizations 2.5 Advanced features 2.6 Alternative methods 3 Analysis 3.1 Convergence 3.2 Consistency and order 3.3 Stability and stiffness 4 History 5 Numerical Solutions to Second Order One Dimensional boundary value problems 6 See also 7 Software for ODE solving 8 References 9 External links Illustration of numerical integration for the differential equation Blue: the Euler method, green: the midpoint method, red: the exact solution, The step size is The problem We want to approximate the solution of the differential equation [edit] where f is a function that maps [t0,∞) × Rd to Rd, and the initial condition y0 ∈ Rd is a given vector. The above formulation is called an initial value problem (IVP). The Picard–Lindelöf theorem states that there is a unique solution, if f is Lipschitz continuous. In contrast, boundary value problems (BVPs) specify (components of) the solution y at more than one point. 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. The same illustration for It is seen that the midpoint method converges faster than the Euler method. Note that we restrict ourselves to first-order differential equations (meaning that only the first derivative of y appears in the equation, and no higher derivatives). This, however, does not restric the generality of the problem, since a higher-order equation can easily be converted to a 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. Methods [edit] Three elementary methods are discussed to give the reader a feeling for the subject. After that, pointers are provided to other methods (which are generally more accurate and efficient). The methods mentioned here are analysed in the next section. The Euler method For more details on this topic, see Euler method. [edit] converted by Web2PDFConvert.com [edit] It is often inefficient to use the same step size all the time. t2. The method is named after Leonhard Euler who described it in 1768. you can find an approximation of a nearby point on the curve by moving a short distance along a line tangent to the curve. this cost must be taken into consideration when one selects the method to use. An extension of this idea is to choose dynamically between different methods of different orders (this is called a variable order method).com . The exponential Euler method If the differential equation is of the form then an approximate explicit solution can be given by [edit] This method is commonly employed in neural simulations and it is the default integrator in the GENESIS neural simulator.. such as the Bulirsch–Stoer algorithm. This means that the methods must also compute an error indicator. converted by Web2PDFConvert. Advanced features A good implementation of one of these methods for solving an ODE entails more than the time-stepping formula. 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. event location: finding the times where.A brief explanation: From any point on a curve. This yields a so-called multistep method. and we construct the sequence t0. Other desirable features include: dense output: cheap numerical approximations for the whole integration interval. t1. The Euler method is an example of an explicit method. Almost all practical multistep methods fall within the family of linear multistep methods. One of their fourth-order methods is especially popular. a particular function vanishes. . but to make the solution depend on more past values. Usually. it costs time to solve this equation. the step size is chosen such that the (local) error per step is below some tolerance level. in contrast with the backward Euler method. named after Carl Runge and Martin Kutta.. which have the form Another possibility is to use more points in the interval [tn. One often uses fixed point iteration or (some modification of) the Newton–Raphson method to achieve this. In more precise terms. Methods based on Richardson extrapolation. This leads to the family of Runge–Kutta methods. it only has order one (the concept of order is explained below). This means that the new value yn+1 is defined in terms of things that are already known.tn+1]. The advantage of implicit methods such as (6) is that they are usually more stable for solving a stiff equation. Of course. t2 = t0 + 2h. say. Motivated by (3). If. to be described below). meaning that a larger step size h can be used. like yn. we use the approximation [edit] 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 possibility is to use not only the previously computed value yn to determine yn+1. This typically requires the use of a root-finding algorithm. We choose a step size h. This caused mathematicians to look for higher-order methods. The backward Euler method For more details on this topic. instead of (2). Perhaps the simplest is the Leapfrog method which is second order and (roughly speaking) relies on two time values. and not only at the points t0. are often used to construct various methods of different orders. … We denote by yn a numerical estimate of the exact solution y(tn). t1 = t0 + h. Rigorous development: Starting with the differential equation (1). so variable step-size methods have been developed.[1] Generalizations [edit] The Euler method is often not accurate enough. we compute these estimates by the following recursive scheme This is the Euler method (or forward Euler method. see Backward Euler method. an estimate of the local error. it may not be the best way to proceed.Leonhard Euler publishes his method. and is often caused by the presence of different time scales in the underlying problem. Consistency is a necessary condition for convergence. [edit] Convergence [edit] A numerical method is said to be convergent if the numerical solution approaches the exact solution as the step size h goes to 0. Some classes of alternative methods are: [edit] multiderivative methods. We said that all higher-order ODEs can be transformed to first-order ODEs of the form (1). we require that for every ODE (1) with a Lipschitz function f and every t* > 0. A related concept is the global (truncation) error.g.. In particular. application of standard methods—such as the Euler method.Augustin Louis Cauchy proves convergence of the Euler method. a numerical scheme has to be convergent to be of any use.. see Stiff equation. so they are consistent. see Truncation error (numerical integration). Most methods being used in practice attain higher order. That is. biology. and stability: whether errors are damped out. and electronics. plasma physics. While this is certainly true.g. or multistep methods (e. 1910 . Stiff problems are ubiquitous in chemical kinetics. which compute the coefficients of the Taylor series of the solution y recursively. Curtiss and Joseph Oakland Hirschfelder coin the term stiff equations. solid mechanics. as well as methods like the Parker–Sochacki method or Bychkov-Scherbakov method. the error sustained in all the steps one needs to reach a fixed time t. 1824 . Cauchy uses the implicit Euler method. More precisely. though other methods may produce stable solutions. control theory. The global error of a pth order one-step method is O(hp). Consistency and order For more details on this topic. it must be both consistent and zero-stable. History Below is a timeline of some important developments in this field.Carl Runge publishes the first Runge–Kutta method. Bashforth. it is the difference between the result given by the method. This "difficult behaviour" in the equation (which may not necessarily be complex itself) is described as stiffness. in particular. Nyström methods work directly with second-order equations. for a method to be convergent. such a method is convergent. order: how well it approximates the solution. Suppose the numerical method is [edit] The local (truncation) error of the method is the error committed by one step of the method. time reversibility Alternative methods Many methods do not fall within the framework discussed here. geometric integration methods are especially designed for special classes of ODEs (e. which use not only the function f but also its derivatives. In fact. methods for second order ODEs. Analysis Numerical analysis is not only the design of numerical methods. Three central concepts in this analysis are: convergence: whether the method approximates the solution. explicit Runge–Kutta methods. Stability and stiffness For more details on this topic. [edit] Numerical Solutions to Second Order One Dimensional boundary value problems [edit] converted by Web2PDFConvert. when used for integrating with respect to time. [edit] For some differential equations. The (forward) Euler method (4) and the backward Euler method (6) introduced above both have order 1. 1952 .Lewis Fry Richardson announces his extrapolation method. the global error at time t is yN − y(t) where N = (t−t0)/h. symplectic integrators for the solution of Hamiltonian equations).com . Adams–Bashforth methods)—exhibit instability in the solutions. Explicitly. They take care that the numerical solution respects the underlying structure or geometry of these classes. assuming that no error was made in earlier steps.Charles F. All the methods mentioned above are convergent.support for parallel computing. This statement is not necessarily true for multi-step methods. 1905 . 1855 . but also their analysis.First mention of the multistep methods of John Couch Adams in a letter written by F. weather forecasting. but not sufficient. Richardson extrapolation. 1768 . 1895 . This class includes Hermite–Obreschkoff methods and Fehlberg methods.Martin Kutta describes the popular fourth-order Runge–Kutta method. In this proof. 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. slightly more demanding than the book by Iserles. Solving ordinary differential equations II: Stiff and differential-algebraic problems. Solving ordinary differential equations I: Nonstiff problems.) v · t · e· First order methods Second order methods Numerical integration Euler method · Backward Euler · Semi-implicit Euler · Exponential Euler Verlet integration · Velocity Verlet · Trapezoidal rule · Beeman's algorithm · Midpoint method · Heun's method · Newmark-beta method · Leapfrog integration [hide] converted by Web2PDFConvert. targeting advanced undergraduate and postgraduate students in mathematics. Application of the Parker–Sochacki Method to Celestial Mechanics . Berlin. Ernst Hairer and Gerhard Wanner. including intermediate steps in the solution. this linear system can then be solved by standard matrix methods. This would lead to equations such as: On first viewing this system of equations appears to have a difficulty associated with the fact that the equation involves no terms that are not multiplied by variables but in fact this is false.g. juin 1996. Butcher. implicit Runge-Kutta. Because see. uses Automatic differentiation.) External links [edit] Joseph W.Denis Diderot. 1991. Berlin.Usually boundary value problems (BVPs) are numerically solved by solving an approximately equivalent matrix problem obtained by discretizing the original BVP.a visual language for differential equation solving Mathematical Assistant on Web online solving first order (linear and with separated variables) and second order linear differential equations (with constant coefficients). Villeneuve d'Ascq : Presses universitaires du Septentrion. ISBN 0-52155376-8 (hardback). Online experiments with JSXGraph Maxima computer algebra system (GPL) References 1.org/BABEL/gum-tutorials/cornelis/doc/html/node9. ^ http://www. ISBN 0-47192990-5. (Extensive online material on ODE numerical analysis history. ISBN 3-540-56670-8. for English-language material on the history of ODE numerical analysis.) Arieh Iserles. Syvert Paul Nørsett and Gerhard Wanner. Rudmin. Dominique Tournès. see e. 1996. ISBN 0-471-96758-0 Ernst Hairer. 1996. John Wiley & Sons. Gear’s BDF and Adams-Moulton). Réimp. 1993. second edition. which also discusses numerical partial differential equations. also can be used online via Sage-server ) odeint . suppose the equation to be solved is: Then the next step could be. C. 468 p.A C++ library for solving ordinary differential equations numerically VisSim . ISBN 0-521-55655-4 (paperback). Springer Verlag. 1997. A First Course in the Numerical Analysis of Differential Equations. For instance. L'intégration approchée des équations différentielles ordinaires (1671-1914) . Chichester.) John Denholm Lambert. (This two-volume monograph systematically covers all aspects of the field. at i=1 and N-1 there's 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 nonhomogenous linear system of equations that has non-trivial solutions See also Courant–Friedrichs–Lewy condition Energy drift List of numerical analysis topics#Numerical methods for ordinary differential equations Reversible reference system propagation algorithm [edit] Software for ODE solving [edit] MATLAB FuncDesigner (free license: BSD. the paper books by Chabert and Goldstine quoted by him. This method in essence uses linear derivative approximations such as where is the distance between neighbouring x values on the discretized domain to obtain a system of linear equations that can be solved. ISBN 3-540-60452-9. Cambridge University Press.com .genesis-sim. 1998. Numerical methods for ordinary differential equations. (Textbook. thèse de doctorat de l'université Paris 7 . Numerical Methods for Ordinary Differential Systems. second edition. The most commonly used method for numerically solving bvps in 1d is called the Finite Difference Method.html [edit] J.NET Initial-value problem for nonstiff and stiff ordinary differential equations (explicit Runge-Kutta. DotNumerics: Ordinary Differential Equations for C# and VB. Springer Verlag. to discretize the problem and use linear derivative approximations such as and solve the resulting system of linear equations. (Textbook. a non-profit organization.Higher order methods Runge–Kutta methods · List of Runge–Kutta methods · Linear multistep method Rate this page What's this? Trustworthy Objective Complete View page ratings Well-written I am highly knowledgeable about this topic (optional) Categories: Numerical differential equations Ordinary differential equations This page was last modified on 25 March 2012 at 21:49. Inc.com . 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