# Runge Kutta methods

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+ | Runge Kutta (RK) methods are an important class of methods for integrating initial value problems formed by [[ODE]]s. Runge Kutta methods encompass a wide selection of numerical methods and some commonly used methods such as Explicit or Implicit [[Euler's Method]], the implicit midpoint rule and the trapezoidal rule are actually simplified versions of a general RK method. | ||

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+ | For the ODE, | ||

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+ | :<math> | ||

+ | y^\prime = f(t,y) | ||

+ | </math> | ||

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+ | the basic idea is to build a series of "stages", <math>k_i</math> that approximate the solution <math>y</math> at various points using samples of <math>f</math> from other stages. Finally, the numerical solution <math>u_{n+1}</math> is constructed from a linear combination of <math>u_n</math> and all the precomputed stages. | ||

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+ | Since the computation of one stage may involve other stages <math>k_i</math> the right hand side <math>f</math> is evaluated in a complicated nonlinear way. The most famous classical RK scheme is described below. | ||

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= Fourth order Runge-Kutta method = | = Fourth order Runge-Kutta method = | ||

## Revision as of 18:27, 23 November 2005

Runge Kutta (RK) methods are an important class of methods for integrating initial value problems formed by ODEs. Runge Kutta methods encompass a wide selection of numerical methods and some commonly used methods such as Explicit or Implicit Euler's Method, the implicit midpoint rule and the trapezoidal rule are actually simplified versions of a general RK method.

For the ODE,

the basic idea is to build a series of "stages", that approximate the solution at various points using samples of from other stages. Finally, the numerical solution is constructed from a linear combination of and all the precomputed stages.

Since the computation of one stage may involve other stages the right hand side is evaluated in a complicated nonlinear way. The most famous classical RK scheme is described below.

# Fourth order Runge-Kutta method

The fourth order Runge-Kutta method could be summarized as:

## Algorithm

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