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Runge–Kutta methods : ウィキペディア英語版
Runge–Kutta methods
In numerical analysis, the Runge–Kutta methods are an important family of implicit and explicit iterative methods, which are used in temporal discretization for the approximation of solutions of ordinary differential equations. These techniques were developed around 1900 by the German mathematicians C. Runge and M. W. Kutta.
See the article on numerical methods for ordinary differential equations for more background and other methods. See also List of Runge–Kutta methods.
==''The'' Runge–Kutta method==
One member of the family of Runge–Kutta methods is often referred to as "RK4", "classical Runge–Kutta method" or simply as "''the'' Runge–Kutta method".
Let an initial value problem be specified as follows.
: \dot = f(t, y), \quad y(t_0) = y_0.
Here, ''y'' is an unknown function (scalar or vector) of time ''t'' which we would like to approximate; we are told that \dot, the rate at which ''y'' changes, is a function of ''t'' and of ''y'' itself. At the initial time t_0 the corresponding ''y''-value is y_0. The function ''f'' and the data t_0, y_0 are given.
Now pick a step-size ''h''>0 and define
:\begin
y_ &= y_n + \tfrac\left(k_1 + 2k_2 + 2k_3 + k_4 \right)\\
t_ &= t_n + h \\
\end
for ''n'' = 0, 1, 2, 3, . . . , using
:
\begin
k_1 &= f(t_n, y_n),
\\
k_2 &= f(t_n + \tfrac, y_n + \tfrac k_1),
\\
k_3 &= f(t_n + \tfrac, y_n + \tfrac k_2),
\\
k_4 &= f(t_n + h, y_n + hk_3).
\end
〔; 〕
:''(Note: the above equations have different but equivalent definitions in different texts).''〔, , and leave out the factor ''h'' in the definition of the stages. , and use the ''y''-values as stages.〕
Here y_ is the RK4 approximation of y(t_), and the next value (y_) is determined by the present value (y_n) plus the weighted average of four increments, where each increment is the product of the size of the interval, ''h'', and an estimated slope specified by function ''f'' on the right-hand side of the differential equation.
* k_1 is the increment based on the slope at the beginning of the interval, using , (Euler's method) ;
* k_2 is the increment based on the slope at the midpoint of the interval, using + \tfrack_1 ;
* k_3 is again the increment based on the slope at the midpoint, but now using + \tfrack_2 ;
* k_4 is the increment based on the slope at the end of the interval, using + hk_3 .
In averaging the four increments, greater weight is given to the increments at the midpoint. If f is independent of y, so that the differential equation is equivalent to a simple integral, then RK4 is Simpson's rule.
The RK4 method is a fourth-order method, meaning that the local truncation error is on the order of O(h^5), while the total accumulated error is order O(h^4).

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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