Lobatto III Runge-Kutta Methods

Lobatto IIIA

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

\[\begin{array}{r|rr} 0 & 0 & 0 \\ 1 & \frac{1}{2} & \frac{1}{2} \\\hline & \frac{1}{2} & \frac{1}{2} \\ \end{array}\]

Runge-Kutta Tableau LobattoIIIA(3) with 3 stages and order 4:

\[\begin{array}{r|rrr} 0 & 0 & 0 & 0 \\ \frac{1}{2} & \frac{5}{24} & \frac{1}{3} & - \frac{1}{24} \\ 1 & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} \\\hline & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} \\ \end{array}\]

Runge-Kutta Tableau LobattoIIIA(4) with 4 stages and order 6:

\[\begin{array}{r|rrrr} 0 & 0 & 0 & 0 & 0 \\ \frac{1}{2} - \frac{\sqrt{5}}{10} & \frac{\sqrt{5}}{120} + \frac{11}{120} & \frac{5}{24} - \frac{\sqrt{5}}{120} & \frac{5}{24} - \frac{13 \sqrt{5}}{120} & - \frac{1}{120} + \frac{\sqrt{5}}{120} \\ \frac{\sqrt{5}}{10} + \frac{1}{2} & \frac{11}{120} - \frac{\sqrt{5}}{120} & \frac{5}{24} + \frac{13 \sqrt{5}}{120} & \frac{\sqrt{5}}{120} + \frac{5}{24} & - \frac{\sqrt{5}}{120} - \frac{1}{120} \\ 1 & \frac{1}{12} & \frac{5}{12} & \frac{5}{12} & \frac{1}{12} \\\hline & \frac{1}{12} & \frac{5}{12} & \frac{5}{12} & \frac{1}{12} \\ \end{array}\]

Lobatto IIIB

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

\[\begin{array}{r|rr} 0 & \frac{1}{2} & 0 \\ 1 & \frac{1}{2} & 0 \\\hline & \frac{1}{2} & \frac{1}{2} \\ \end{array}\]

Runge-Kutta Tableau LobattoIIIB(3) with 3 stages and order 4:

\[\begin{array}{r|rrr} 0 & \frac{1}{6} & - \frac{1}{6} & 0 \\ \frac{1}{2} & \frac{1}{6} & \frac{1}{3} & 0 \\ 1 & \frac{1}{6} & \frac{5}{6} & 0 \\\hline & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} \\ \end{array}\]

Runge-Kutta Tableau LobattoIIIB(4) with 4 stages and order 6:

\[\begin{array}{r|rrrr} 0 & \frac{1}{12} & - \frac{\sqrt{5}}{24} - \frac{1}{24} & - \frac{1}{24} + \frac{\sqrt{5}}{24} & 0 \\ \frac{1}{2} - \frac{\sqrt{5}}{10} & \frac{1}{12} & \frac{\sqrt{5}}{120} + \frac{5}{24} & \frac{5}{24} - \frac{13 \sqrt{5}}{120} & 0 \\ \frac{\sqrt{5}}{10} + \frac{1}{2} & \frac{1}{12} & \frac{5}{24} + \frac{13 \sqrt{5}}{120} & \frac{5}{24} - \frac{\sqrt{5}}{120} & 0 \\ 1 & \frac{1}{12} & \frac{11}{24} - \frac{\sqrt{5}}{24} & \frac{\sqrt{5}}{24} + \frac{11}{24} & 0 \\\hline & \frac{1}{12} & \frac{5}{12} & \frac{5}{12} & \frac{1}{12} \\ \end{array}\]

Lobatto IIIC

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

\[\begin{array}{r|rr} 0 & \frac{1}{2} & - \frac{1}{2} \\ 1 & \frac{1}{2} & \frac{1}{2} \\\hline & \frac{1}{2} & \frac{1}{2} \\ \end{array}\]

Runge-Kutta Tableau LobattoIIIC(3) with 3 stages and order 4:

\[\begin{array}{r|rrr} 0 & \frac{1}{6} & - \frac{1}{3} & \frac{1}{6} \\ \frac{1}{2} & \frac{1}{6} & \frac{5}{12} & - \frac{1}{12} \\ 1 & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} \\\hline & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} \\ \end{array}\]

Runge-Kutta Tableau LobattoIIIC(4) with 4 stages and order 6:

\[\begin{array}{r|rrrr} 0 & \frac{1}{12} & - \frac{\sqrt{5}}{12} & \frac{\sqrt{5}}{12} & - \frac{1}{12} \\ \frac{1}{2} - \frac{\sqrt{5}}{10} & \frac{1}{12} & \frac{1}{4} & \frac{1}{6} - \frac{7 \sqrt{5}}{60} & \frac{\sqrt{5}}{60} \\ \frac{\sqrt{5}}{10} + \frac{1}{2} & \frac{1}{12} & \frac{1}{6} + \frac{7 \sqrt{5}}{60} & \frac{1}{4} & - \frac{\sqrt{5}}{60} \\ 1 & \frac{1}{12} & \frac{5}{12} & \frac{5}{12} & \frac{1}{12} \\\hline & \frac{1}{12} & \frac{5}{12} & \frac{5}{12} & \frac{1}{12} \\ \end{array}\]

Lobatto IIIC̄

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

\[\begin{array}{r|rr} 0 & 0 & 0 \\ 1 & 1 & 0 \\\hline & \frac{1}{2} & \frac{1}{2} \\ \end{array}\]

Runge-Kutta Tableau LobattoIIIC̄(3) with 3 stages and order 4:

\[\begin{array}{r|rrr} 0 & 0 & 0 & 0 \\ \frac{1}{2} & \frac{1}{4} & \frac{1}{4} & 0 \\ 1 & 0 & 1 & 0 \\\hline & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} \\ \end{array}\]

Runge-Kutta Tableau LobattoIIIC̄(4) with 4 stages and order 6:

\[\begin{array}{r|rrrr} 0 & 0 & 0 & 0 & 0 \\ \frac{1}{2} - \frac{\sqrt{5}}{10} & \frac{\sqrt{5}}{60} + \frac{1}{12} & \frac{1}{6} & \frac{1}{4} - \frac{7 \sqrt{5}}{60} & 0 \\ \frac{\sqrt{5}}{10} + \frac{1}{2} & \frac{1}{12} - \frac{\sqrt{5}}{60} & \frac{1}{4} + \frac{7 \sqrt{5}}{60} & \frac{1}{6} & 0 \\ 1 & \frac{1}{6} & \frac{5}{12} - \frac{\sqrt{5}}{12} & \frac{\sqrt{5}}{12} + \frac{5}{12} & 0 \\\hline & \frac{1}{12} & \frac{5}{12} & \frac{5}{12} & \frac{1}{12} \\ \end{array}\]

Lobatto IIID

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

\[\begin{array}{r|rr} 0 & \frac{1}{4} & - \frac{1}{4} \\ 1 & \frac{3}{4} & \frac{1}{4} \\\hline & \frac{1}{2} & \frac{1}{2} \\ \end{array}\]

Runge-Kutta Tableau LobattoIIID(3) with 3 stages and order 4:

\[\begin{array}{r|rrr} 0 & \frac{1}{12} & - \frac{1}{6} & \frac{1}{12} \\ \frac{1}{2} & \frac{5}{24} & \frac{1}{3} & - \frac{1}{24} \\ 1 & \frac{1}{12} & \frac{5}{6} & \frac{1}{12} \\\hline & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} \\ \end{array}\]

Runge-Kutta Tableau LobattoIIID(4) with 4 stages and order 6:

\[\begin{array}{r|rrrr} 0 & \frac{1}{24} & - \frac{\sqrt{5}}{24} & \frac{\sqrt{5}}{24} & - \frac{1}{24} \\ \frac{1}{2} - \frac{\sqrt{5}}{10} & \frac{\sqrt{5}}{120} + \frac{1}{12} & \frac{5}{24} & \frac{5}{24} - \frac{7 \sqrt{5}}{60} & \frac{\sqrt{5}}{120} \\ \frac{\sqrt{5}}{10} + \frac{1}{2} & \frac{1}{12} - \frac{\sqrt{5}}{120} & \frac{5}{24} + \frac{7 \sqrt{5}}{60} & \frac{5}{24} & - \frac{\sqrt{5}}{120} \\ 1 & \frac{1}{8} & \frac{5}{12} - \frac{\sqrt{5}}{24} & \frac{\sqrt{5}}{24} + \frac{5}{12} & \frac{1}{24} \\\hline & \frac{1}{12} & \frac{5}{12} & \frac{5}{12} & \frac{1}{12} \\ \end{array}\]

Lobatto IIIE

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

\[\begin{array}{r|rr} 0 & \frac{1}{4} & 0 \\ 1 & \frac{1}{2} & \frac{1}{4} \\\hline & \frac{1}{2} & \frac{1}{2} \\ \end{array}\]

Runge-Kutta Tableau LobattoIIIE(3) with 3 stages and order 4:

\[\begin{array}{r|rrr} 0 & \frac{1}{12} & - \frac{1}{12} & 0 \\ \frac{1}{2} & \frac{3}{16} & \frac{1}{3} & - \frac{1}{48} \\ 1 & \frac{1}{6} & \frac{3}{4} & \frac{1}{12} \\\hline & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} \\ \end{array}\]

Runge-Kutta Tableau LobattoIIIE(4) with 4 stages and order 6:

\[\begin{array}{r|rrrr} 0 & \frac{1}{24} & - \frac{\sqrt{5}}{48} - \frac{1}{48} & - \frac{1}{48} + \frac{\sqrt{5}}{48} & 0 \\ \frac{1}{2} - \frac{\sqrt{5}}{10} & \frac{\sqrt{5}}{240} + \frac{7}{80} & \frac{5}{24} & \frac{5}{24} - \frac{13 \sqrt{5}}{120} & - \frac{1}{240} + \frac{\sqrt{5}}{240} \\ \frac{\sqrt{5}}{10} + \frac{1}{2} & \frac{7}{80} - \frac{\sqrt{5}}{240} & \frac{5}{24} + \frac{13 \sqrt{5}}{120} & \frac{5}{24} & - \frac{\sqrt{5}}{240} - \frac{1}{240} \\ 1 & \frac{1}{12} & \frac{7}{16} - \frac{\sqrt{5}}{48} & \frac{\sqrt{5}}{48} + \frac{7}{16} & \frac{1}{24} \\\hline & \frac{1}{12} & \frac{5}{12} & \frac{5}{12} & \frac{1}{12} \\ \end{array}\]

Lobatto IIIF

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

\[\begin{array}{r|rr} 0 & \frac{1}{12} & - \frac{1}{12} \\ 1 & \frac{7}{12} & \frac{5}{12} \\\hline & \frac{1}{2} & \frac{1}{2} \\ \end{array}\]

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

\[\begin{array}{r|rrr} 0 & \frac{1}{30} & - \frac{1}{15} & \frac{1}{30} \\ \frac{1}{2} & \frac{5}{24} & \frac{1}{3} & - \frac{1}{24} \\ 1 & \frac{2}{15} & \frac{11}{15} & \frac{2}{15} \\\hline & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} \\ \end{array}\]

Runge-Kutta Tableau LobattoIIIF(4) with 4 stages and order 8:

\[\begin{array}{r|rrrr} 0 & \frac{1}{56} & - \frac{\sqrt{5}}{56} & \frac{\sqrt{5}}{56} & - \frac{1}{56} \\ \frac{1}{2} - \frac{\sqrt{5}}{10} & \frac{\sqrt{5}}{120} + \frac{37}{420} & \frac{5}{24} - \frac{\sqrt{5}}{210} & \frac{5}{24} - \frac{47 \sqrt{5}}{420} & - \frac{1}{210} + \frac{\sqrt{5}}{120} \\ \frac{\sqrt{5}}{10} + \frac{1}{2} & \frac{37}{420} - \frac{\sqrt{5}}{120} & \frac{5}{24} + \frac{47 \sqrt{5}}{420} & \frac{\sqrt{5}}{210} + \frac{5}{24} & - \frac{\sqrt{5}}{120} - \frac{1}{210} \\ 1 & \frac{17}{168} & \frac{5}{12} - \frac{\sqrt{5}}{56} & \frac{\sqrt{5}}{56} + \frac{5}{12} & \frac{11}{168} \\\hline & \frac{1}{12} & \frac{5}{12} & \frac{5}{12} & \frac{1}{12} \\ \end{array}\]

Lobatto IIIG

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

\[\begin{array}{r|rr} 0 & \frac{1}{4} & - \frac{1}{12} \\ 1 & \frac{7}{12} & \frac{1}{4} \\\hline & \frac{1}{2} & \frac{1}{2} \\ \end{array}\]

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

\[\begin{array}{r|rrr} 0 & \frac{1}{12} & - \frac{7}{60} & \frac{1}{30} \\ \frac{1}{2} & \frac{47}{240} & \frac{1}{3} & - \frac{7}{240} \\ 1 & \frac{2}{15} & \frac{47}{60} & \frac{1}{12} \\\hline & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} \\ \end{array}\]

Runge-Kutta Tableau LobattoIIIG(4) with 4 stages and order 8:

\[\begin{array}{r|rrrr} 0 & \frac{1}{24} & - \frac{5 \sqrt{5}}{168} - \frac{1}{84} & - \frac{1}{84} + \frac{5 \sqrt{5}}{168} & - \frac{1}{56} \\ \frac{1}{2} - \frac{\sqrt{5}}{10} & \frac{\sqrt{5}}{168} + \frac{3}{35} & \frac{5}{24} & \frac{5}{24} - \frac{47 \sqrt{5}}{420} & - \frac{1}{420} + \frac{\sqrt{5}}{168} \\ \frac{\sqrt{5}}{10} + \frac{1}{2} & \frac{3}{35} - \frac{\sqrt{5}}{168} & \frac{5}{24} + \frac{47 \sqrt{5}}{420} & \frac{5}{24} & - \frac{\sqrt{5}}{168} - \frac{1}{420} \\ 1 & \frac{17}{168} & \frac{3}{7} - \frac{5 \sqrt{5}}{168} & \frac{5 \sqrt{5}}{168} + \frac{3}{7} & \frac{1}{24} \\\hline & \frac{1}{12} & \frac{5}{12} & \frac{5}{12} & \frac{1}{12} \\ \end{array}\]