Editing Runge–Kutta methods (section)

==Runge–Kutta–Nyström methods==

Runge–Kutta–Nyström methods are specialized Runge–Kutta methods that are optimized for second-order differential equations.<ref>{{cite journal |last1=Dormand |first1=J. R. |last2=Prince |first2=P. J. |title=New Runge–Kutta Algorithms for Numerical Simulation in Dynamical Astronomy |journal=Celestial Mechanics |date=October 1978 |volume=18 |issue=3 |pages=223–232|doi=10.1007/BF01230162 |bibcode=1978CeMec..18..223D |s2cid=120974351 }}</ref><ref>{{cite report | last=Fehlberg | first=E. | date = October 1974 | title = Classical seventh-, sixth-, and fifth-order Runge–Kutta–Nyström formulas with stepsize control for general second-order differential equations | publisher   = National Aeronautics and Space Administration | edition =NASA TR R-432 | location    =Marshall Space Flight Center, AL
}}</ref> A general Runge–Kutta–Nyström method for a second-order ODE system

<math display=block>
\ddot y_i = f_i (y_1, y_2, \ldots, y_n )
</math>

with order <math>s</math> is with the form 

<math display=block>\begin{cases}
g_i = y_m + c_i h {\dot y}_m + h^2 \sum_{j=1}^s a_{ij} f(g_j), & i=1,2,\ldots, s\\
y_{m+1} = y_{m} + h {\dot y}_m + h^2 \sum_{j=1}^s \bar{b}_j f(g_j) \\
{\dot y}_{m+1} = {\dot y}_m + h \sum_{j=1}^s b_j f(g_j)
\end{cases}</math>

which forms a Butcher table with the form

<math display=block>
\begin{array}{c|cccc}
c_1    & a_{11} & a_{12}& \dots & a_{1s}\\
c_2    & a_{21} & a_{22}& \dots & a_{2s}\\
\vdots & \vdots & \vdots& \ddots& \vdots\\
c_s    & a_{s1} & a_{s2}& \dots & a_{ss} \\
\hline
       & {\bar b}_1    & {\bar b}_2   & \dots & {\bar b}_s\\
       & b_1  & b_2 & \dots & b_s
\end{array} =
\begin{array}{c|c}
\mathbf{c}& \mathbf{A}\\
\hline
          & \mathbf{{\bar b}}^\top \\
          & \mathbf{b}^\top
\end{array}
</math>

Two fourth-order explicit RKN methods are given by the following Butcher tables:

<math display=block>
\begin{array}{c|ccc}
c_i & & a_{ij}  & \\
\frac{3+\sqrt{3}}{6} & 0 & 0 & 0 \\
\frac{3-\sqrt{3}}{6} & \frac{2-\sqrt{3}}{12} & 0 & 0 \\
\frac{3+\sqrt{3}}{6} & 0 & \frac{\sqrt{3}}{6} & 0 \\
\hline
\overline{b_i} & \frac{5-3 \sqrt{3}}{24} & \frac{3+\sqrt{3}}{12} & \frac{1+\sqrt{3}}{24} \\
\hline
b_i & \frac{3-2 \sqrt{3}}{12} & \frac{1}{2} & \frac{3+2 \sqrt{3}}{12}
\end{array}
</math>
<math display=block>
\begin{array}{c|ccc}
c_i & & a_{ij}  & \\
\frac{3-\sqrt{3}}{6} & 0 & 0 & 0 \\
\frac{3+\sqrt{3}}{6} & \frac{2+\sqrt{3}}{12} & 0 & 0 \\
\frac{3-\sqrt{3}}{6} & 0 & -\frac{\sqrt{3}}{6} & 0 \\
\hline
\overline{b_i} & \frac{5+3 \sqrt{3}}{24} & \frac{3-\sqrt{3}}{12} & \frac{1-\sqrt{3}}{24} \\
\hline
b_i & \frac{3+2 \sqrt{3}}{12} & \frac{1}{2} & \frac{3-2 \sqrt{3}}{12}
\end{array}
</math>

These two schemes also have the symplectic-preserving properties when the original equation is derived from a conservative classical mechanical system, i.e. when 

<math display=block>
f_i(x_1, \ldots, x_n ) = \frac{\partial V}{\partial x_i} (x_1, \ldots, x_n )
</math>

for some scalar function <math> V </math>. <ref>{{Cite journal |last1=Qin |first1=Meng-Zhao |last2=Zhu |first2=Wen-Jie |date=1991-01-01 |title=Canonical Runge-Kutta-Nyström (RKN) methods for second order ordinary differential equations |url=https://dx.doi.org/10.1016/0898-1221%2891%2990209-M |journal=Computers & Mathematics with Applications |volume=22 |issue=9 |pages=85–95 |doi=10.1016/0898-1221(91)90209-M |issn=0898-1221}}</ref>