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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". 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.

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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". 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. In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.

A first-order differential equation is an Initial value problem IVP of the form, [2]. 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. 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 , [3] Galerkin methods , [4] or collocation methods are appropriate for that class of problems.

Numerical methods for solving first-order IVPs often fall into one of two large categories: [5] 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. 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. The so-called general linear methods GLMs are a generalization of the above two large classes of methods.

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.

This formula is usually applied in the following way. 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 The Euler method is an example of an explicit method. 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.

Exponential integrators describe a large class of integrators that have recently seen a lot of development. 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. 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. One of their fourth-order methods is especially popular.

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 , [14] such as the Bulirsch—Stoer algorithm , [15] [16] are often used to construct various methods of different orders. Many methods do not fall within the framework discussed here. Some classes of alternative methods are:. For applications that require parallel computing on supercomputers , the degree of concurrency offered by a numerical method becomes relevant.

In view of the challenges from exascale computing systems, numerical methods for initial value problems which can provide concurrency in temporal direction are being studied. Numerical analysis is not only the design of numerical methods, but also their analysis. Three central concepts in this analysis are:. A numerical method is said to be convergent if the numerical solution approaches the exact solution as the step size h goes to 0. 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:. Hence a method is consistent if it has an order greater than 0.

The forward Euler method 4 and the backward Euler method 6 introduced above both have order 1, so they are consistent. Most methods being used in practice attain higher order. Consistency is a necessary condition for convergence [ citation needed ] , 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.

The global error of a p th order one-step method is O h p ; in particular, such a method is convergent. This statement is not necessarily true for multi-step methods. For some differential equations, application of standard methods—such as the Euler method, explicit Runge—Kutta methods , or multistep methods for example, Adams—Bashforth methods —exhibit instability in the solutions, though other methods may produce stable solutions.

This "difficult behaviour" in the equation which may not necessarily be complex itself is described as stiffness , and is often caused by the presence of different time scales in the underlying problem. Stiff problems are ubiquitous in chemical kinetics , control theory , solid mechanics , weather forecasting , biology , plasma physics , and electronics.

One way to overcome stiffness is to extend the notion of differential equation to that of differential inclusion , which allows for and models non-smoothness.

Below is a timeline of some important developments in this field. Boundary value problems BVPs are usually solved numerically by solving an approximately equivalent matrix problem obtained by discretizing the original BVP. For example, the second-order central difference approximation to the first derivative is given by:.

One then constructs a linear system that can then be solved by standard matrix methods. For example, suppose the equation to be solved is:.

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

From Wikipedia, the free encyclopedia. Further information: Euler method. Further information: Backward Euler method. Further information: Exponential integrator. Main articles: Sequence , Limit mathematics , and Limit of a sequence. Further information: Truncation error numerical integration. Further information: Stiff equation. Ordinary differential equations with applications Vol. Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems Vol.

Scholarpedia, 5 10 Numerical methods for ordinary differential equations: initial value problems. Diagonally implicit Runge-Kutta formulae with error estimates. Strong stability of singly-diagonally-implicit Runge—Kutta methods. Applied Numerical Mathematics, 58 11 , An efficient integrator that uses Gauss-Radau spacings. Cambridge University Press. The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods.

Extrapolation methods: theory and practice. Extrapolation and the Bulirsch-Stoer algorithm. Physical Review E, 65 6 , Implementation of the Bulirsch Stoer extrapolation method. Parker-Sochacki method for solving systems of ordinary differential equations using graphics processors.

Numerical Analysis and Applications, 4 3 , Geometric numerical integration: structure-preserving algorithms for ordinary differential equations Vol. Acta Numerica, 12, Contributions in Mathematical and Computational Sciences.

Springer International Publishing. Communications of the ACM. Accuracy and stability of numerical algorithms Vol.

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