Gauss-Legendre Runge-Kutta Methods

Runge-Kutta Tableau Gauss(1) with 1 stages and order 2:

\[\begin{array}{r|r} \frac{1}{2} & \frac{1}{2} \\\hline & 1 \\ \end{array}\]

Runge-Kutta Tableau Gauss(2) with 2 stages and order 4:

\[\begin{array}{r|rr} \frac{1}{2} - \frac{\sqrt{3}}{6} & \frac{1}{4} & \frac{1}{4} - \frac{\sqrt{3}}{6} \\ \frac{\sqrt{3}}{6} + \frac{1}{2} & \frac{1}{4} + \frac{\sqrt{3}}{6} & \frac{1}{4} \\\hline & \frac{1}{2} & \frac{1}{2} \\ \end{array}\]

Runge-Kutta Tableau Gauss(3) with 3 stages and order 6:

\[\begin{array}{r|rrr} \frac{1}{2} - \frac{\sqrt{15}}{10} & \frac{5}{36} & \frac{2}{9} - \frac{\sqrt{15}}{15} & \frac{5}{36} - \frac{\sqrt{15}}{30} \\ \frac{1}{2} & \frac{5}{36} + \frac{\sqrt{15}}{24} & \frac{2}{9} & \frac{5}{36} - \frac{\sqrt{15}}{24} \\ \frac{\sqrt{15}}{10} + \frac{1}{2} & \frac{\sqrt{15}}{30} + \frac{5}{36} & \frac{2}{9} + \frac{\sqrt{15}}{15} & \frac{5}{36} \\\hline & \frac{5}{18} & \frac{4}{9} & \frac{5}{18} \\ \end{array}\]