# The first sixth order methods

The order conditions for a Runge–Kutta method with order $p$ and $s$ stages are a complicated system of polynomial equations up to degree $p$, in $s(s+1)/2$ variables. Above $p=4$, these equations differ for a scalar differential equation and a higher dimensional vector equation. Some details are shown in Table 1.

**Table 1**

- If he had known that $37$ conditions had to be solved for a differential equation system of arbitrary dimension, he would have chosen $s=9$ .
- But, surprisingly, his methods actually satisfy the additional $6$ conditions.
- Perhaps even more surprisingly, sixth order methods exist which satisfy the $37$ conditions for a high dimensional problem with only $s=7$ stages. This means that a system of $37$ equations in $28$ uknowns has solutions.

In [J.C. Butcher, 1964] new sixth order methods were derived with $s=7$ stages. In this set of methods, a number of assumptions were made to simplify the derivation. These were \[ \begin{array}{r l r l} \sum_{i=1}^7 b_i a_{ij} &\h= b_j(1-c_j), & j=1,\dots,7,\\ \sum_{i=1}^7 a_{ij} c_j&\h= \frac12 c_i^2, & i =3,\dots,7,\\ b_2 &\h= 0,\\ \sum_{i>2} b_i c_i a_{i2}&\h= 0,\\ \sum_{i>2} b_i c_i^2 a_{i2}&\h= 0,\\ \sum_{i>j>2} b_i c_i a_{ij} a_{j2}&\h= 0. \end{array} \] These assumptions can lead to a sixth order method only if \[ c_4 = \frac{c_3}{2-10c_3+15c_3^2}. \] From the methods that arise from these developments, two representative examples are \[ \begin{array}{c|ccccccc} 0 \g&\g\\ \frac13 \g&\g \frac13 \g&\g\\ \frac23 \g&\g 0 \g&\g \frac23 \g&\g\\ \frac13\g&\g\frac1{12} \g&\g \frac13 \g&\g-\frac1{12}\m \g&\g\\ \frac12 \g&\g-\frac1{16}\m \g&\g\frac98 \g&\g -\frac3{16}\m \g&\g -\frac38\m \g&\g\\ \frac12\g&\g0\g&\g\frac98\g&\g-\frac38\m\g&\g-\frac34\m\g&\g\frac12\g&\g\\ 1 \g&\g \frac9{44} \g&\g-\frac9{11} \m\g&\g\frac{63}{44} \g&\g\frac{18}{11} \g&\g 0\g&\g -\frac{16}{11} \m \g&\g\\ \hline \g&\g\frac{11}{120} \g&\g 0 \g&\g \frac{27}{40} \g&\g \frac{27}{40} \g&\g-\frac4{15}\m\g&\g -\frac4{15}\m\g&\g\frac{11}{120} \end{array} \] and \[ \begin{array}{c|ccccccc} 0 \C\\ \frac{1}{2} \C \frac{1}{2}\C\\ \frac{2}{3} \C \frac{2}{9} \C \frac{4}{9}\C\\ \frac{1}{3} \C \frac{7}{36} \C \frac{2}{9} \C -\frac{1}{12}\s\C\\ \frac{5}{6} \C -\frac{35}{144}\s \C -\frac{55}{36}\s \C \frac{35}{48} \C \frac{15}{8}\C\\ \frac{1}{6} \C -\frac{1}{360}\s \C -\frac{11}{36}\s \C -\frac{1}{8}\s \C \frac{1}{2} \C \frac{1}{10}\C\\ 1 \C -\frac{41}{260}\s \C \frac{22}{13} \C \frac{43}{156} \C -\frac{118}{39}\s \C \frac{32}{195} \C\;\; \frac{80}{39}\C\\ \hline \C \frac{13}{200} \C 0 \C \frac{11}{40} \C \frac{11}{40} \C \frac{4}{25} \C \;\;\frac{4}{25} \C\;\: \frac{13}{200} \end{array} \] In addition to methods with rational coefficients, there exist methods based on Lobatto quadrature, such as the following from

**References**

**3**, 185–201.

**4**, 179–194.

**9**, 389–405.

**1**, 201–224.

**2**, 21–24.

$\to$ The birth of B-series

$\to$ The first eighth order methods

$\to$ B-series book

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