“Runge–Kutta methods”的概念、定义、翻译、参考文献-科学参考

Runge–Kutta methods

Mathematics

A numerical method of solving differential equations in the form $\frac{d y}{d x} = f (x, y)$ which uses the midpoint of interval(s) to improve accuracy. So if the value of the function is known at (x_n,y_n), and the estimate y_n + 1 of y is required at x_n + 1 = x_n + h, the second-order Runge–Kutta formula is

$\begin{array}{l} k_{1} = h \times f (x_{n}, y_{n}) \\ k_{2} = h \times f (x_{n} + \frac{1}{2} h, y_{n} + \frac{1}{2} k_{1}) \\ y_{n + 1} = y_{n} + k_{2} \end{array}$

and the fourth-order Runge–Kutta formula is

$\begin{array}{l} k_{1} = h \times f (x_{n}, y_{n}) \\ k_{2} = h \times f (x_{n} + \frac{1}{2} h, y_{n} + \frac{1}{2} k_{1}) \\ k_{3} = h \times f (x_{n} + \frac{1}{2} h, y_{n} + \frac{1}{2} k_{2}) \\ k_{4} = h \times f (x_{n} + h, y_{n} + k_{3}) \\ y_{n + 1} = y_{n} + \frac{1}{6} k_{1} + \frac{1}{3} k_{2} + \frac{1}{3} k_{3} + \frac{1}{6} k_{4} \end{array}$

Computer

A widely used class of methods for the numerical solution of ordinary differential equations. For the initial-value problem
$y' = f (x, y), y (x_{0}) = y_{0},$
the general form of the m-stage method is
$\begin{array}{l} k_{i} = f (x_{n} + c_{i} h, y_{n} + h Σ_{j = 1}^{m} a_{i j} k_{j}) \\ i = 1, 2, \dots, m \\ y_{n + 1} = y_{n} + h Σ_{i = 1}^{m} b_{i} k_{i}) \\ x_{n + 1} = x_{n} + h \end{array}$
The derivation of suitable parameters a_ij, b_i, and c_i requires extremely lengthy algebraic manipulations, except for small values of m.
Some early examples were developed by Runge and a systematic treatment was initiated by Kutta about 1900. Recently, significant advances have been made in the development of a general theory and in the derivation and implementation of efficient methods incorporating error estimation and control.
Except for stiff equations (see ordinary differential equations), explicit methods
$with a_{i j} = 0, j \geq i$
are used. These are relatively easy to program and are efficient compared with other methods unless evaluations of f(x,y) are expensive.
To be useful for practical problems, the methods should be implemented in a form that allows the stepsize h to vary across the range of integration. Methods for choosing the steps h are based on estimates of the local error. A Runge–Kutta formula should also be derived with a local interpolant that can be used to produce accurate approximations for all values of x, not just at the grid-points x_n. This avoids the considerable extra cost caused by artificially restricting the stepsize when dense output is required.

单词	Runge–Kutta methods
释义	Runge–Kutta methods Mathematics A numerical method of solving differential equations in the form $\frac{d y}{d x} = f (x, y)$ which uses the midpoint of interval(s) to improve accuracy. So if the value of the function is known at (x_n,y_n), and the estimate y_n + 1 of y is required at x_n + 1 = x_n + h, the second-order Runge–Kutta formula is $\begin{array}{l} k_{1} = h \times f (x_{n}, y_{n}) \\ k_{2} = h \times f (x_{n} + \frac{1}{2} h, y_{n} + \frac{1}{2} k_{1}) \\ y_{n + 1} = y_{n} + k_{2} \end{array}$ and the fourth-order Runge–Kutta formula is $\begin{array}{l} k_{1} = h \times f (x_{n}, y_{n}) \\ k_{2} = h \times f (x_{n} + \frac{1}{2} h, y_{n} + \frac{1}{2} k_{1}) \\ k_{3} = h \times f (x_{n} + \frac{1}{2} h, y_{n} + \frac{1}{2} k_{2}) \\ k_{4} = h \times f (x_{n} + h, y_{n} + k_{3}) \\ y_{n + 1} = y_{n} + \frac{1}{6} k_{1} + \frac{1}{3} k_{2} + \frac{1}{3} k_{3} + \frac{1}{6} k_{4} \end{array}$ Computer A widely used class of methods for the numerical solution of ordinary differential equations. For the initial-value problem $y' = f (x, y), y (x_{0}) = y_{0},$ the general form of the m-stage method is $\begin{array}{l} k_{i} = f (x_{n} + c_{i} h, y_{n} + h Σ_{j = 1}^{m} a_{i j} k_{j}) \\ i = 1, 2, \dots, m \\ y_{n + 1} = y_{n} + h Σ_{i = 1}^{m} b_{i} k_{i}) \\ x_{n + 1} = x_{n} + h \end{array}$ The derivation of suitable parameters a_ij, b_i, and c_i requires extremely lengthy algebraic manipulations, except for small values of m. Some early examples were developed by Runge and a systematic treatment was initiated by Kutta about 1900. Recently, significant advances have been made in the development of a general theory and in the derivation and implementation of efficient methods incorporating error estimation and control. Except for stiff equations (see ordinary differential equations), explicit methods $with a_{i j} = 0, j \geq i$ are used. These are relatively easy to program and are efficient compared with other methods unless evaluations of f(x,y) are expensive. To be useful for practical problems, the methods should be implemented in a form that allows the stepsize h to vary across the range of integration. Methods for choosing the steps h are based on estimates of the local error. A Runge–Kutta formula should also be derived with a local interpolant that can be used to produce accurate approximations for all values of x, not just at the grid-points x_n. This avoids the considerable extra cost caused by artificially restricting the stepsize when dense output is required.
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