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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>
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y^\prime = f(t,y)
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</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,


y^\prime = f(t,y)

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

Since the computation of one stage may involve other stages k_i the right hand side f 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

\dot y = f\left( {x,y} \right)
k_1  = hf\left( {x_n ,y_n } \right)
k_2  = hf\left( {x_n  + {h \over 2},y_n  + {{k_1 } \over 2}} \right)
k_3  = hf\left( {x_n  + {h \over 2},y_n  + {{k_2 } \over 2}} \right)
k_4  = hf\left( {x_n  + h,y_n  + k_3 } \right)
y_{n + 1}  = y_n  + {{k_1 } \over 6} + {{k_2 } \over 3} + {{k_3 } \over 3} + {{k_4 } \over 6}




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