Runge Kutta methods
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==Algorithm== | ==Algorithm== | ||
| - | ::<math> | + | ::<math>y^\prime = f\left( {t,y} \right) </math> |
| - | ::<math>k_1 = hf\left( { | + | ::<math>k_1 = hf\left( {t_n ,y_n } \right) </math> |
| - | ::<math>k_2 = hf\left( { | + | ::<math>k_2 = hf\left( {t_n + {h \over 2},y_n + {{k_1 } \over 2}} \right) </math> |
| - | ::<math>k_3 = hf\left( { | + | ::<math>k_3 = hf\left( {t_n + {h \over 2},y_n + {{k_2 } \over 2}} \right) </math> |
| - | ::<math>k_4 = hf\left( { | + | ::<math>k_4 = hf\left( {t_n + h,y_n + k_3 } \right) </math> |
::<math>y_{n + 1} = y_n + {{k_1 } \over 6} + {{k_2 } \over 3} + {{k_3 } \over 3} + {{k_4 } \over 6} </math> | ::<math>y_{n + 1} = y_n + {{k_1 } \over 6} + {{k_2 } \over 3} + {{k_3 } \over 3} + {{k_4 } \over 6} </math> | ||
Revision as of 18:29, 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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