List of Runge–Kutta methods

Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation

${\displaystyle {\frac {dy}{dt}}=f(t,y).}$

Explicit Runge–Kutta methods take the form

${\displaystyle {\begin{aligned}y_{n+1}&=y_{n}+h\sum _{i=1}^{s}b_{i}k_{i}\\k_{1}&=f(t_{n},y_{n}),\\k_{2}&=f(t_{n}+c_{2}h,y_{n}+h(a_{21}k_{1})),\\k_{3}&=f(t_{n}+c_{3}h,y_{n}+h(a_{31}k_{1}+a_{32}k_{2})),\\&\;\;\vdots \\k_{i}&=f\left(t_{n}+c_{i}h,y_{n}+h\sum _{j=1}^{i-1}a_{ij}k_{j}\right).\end{aligned}}}$

Stages for implicit methods of s stages take the more general form, with the solution to be found over all s

${\displaystyle k_{i}=f\left(t_{n}+c_{i}h,y_{n}+h\sum _{j=1}^{s}a_{ij}k_{j}\right).}$

Each method listed on this page is defined by its Butcher tableau, which puts the coefficients of the method in a table as follows:

${\displaystyle {\begin{array}{c|cccc}c_{1}&a_{11}&a_{12}&\dots &a_{1s}\\c_{2}&a_{21}&a_{22}&\dots &a_{2s}\\\vdots &\vdots &\vdots &\ddots &\vdots \\c_{s}&a_{s1}&a_{s2}&\dots &a_{ss}\\\hline &b_{1}&b_{2}&\dots &b_{s}\\\end{array}}}$

For adaptive and implicit methods, the Butcher tableau is extended to give values of ${\displaystyle b_{i}^{*}}$, and the estimated error is then

${\displaystyle e_{n+1}=h\sum _{i=1}^{s}(b_{i}-b_{i}^{*})k_{i}}$.