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List of Runge–Kutta methods
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<p> <a href="/facts/Runge%E2%80%93Kutta_methods/YRjbyHpY">Runge–Kutta methods</a> are methods for the numerical solution of the <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equation</a> </p> d y d t = f ( t , y ) . {\displaystyle {\frac {dy}{dt}}=f(t,y).} <p><a href="/facts/Explicit_and_implicit_methods/jnJhWfuh">Explicit</a> Runge–Kutta methods take the form </p> y n + 1 = y n + h ∑ 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 ) ) , ⋮ k i = f ( t n + c i h , y n + h ∑ j = 1 i − 1 a i j k j ) . {\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}}} <p>Stages for <a href="/facts/Explicit_and_implicit_methods/jnJhWfuh">implicit methods</a> of s stages take the more general form, with the <a href="/facts/Explicit_and_implicit_methods/jnJhWfuh">solution to be found</a> over all s </p> k i = f ( t n + c i h , y n + h ∑ j = 1 s a i j k j ) . {\displaystyle k_{i}=f\left(t_{n}+c_{i}h,y_{n}+h\sum _{j=1}^{s}a_{ij}k_{j}\right).} <p>Each method listed on this page is defined by its <a href="/facts/Butcher_tableau/YRjbyHpY">Butcher tableau</a>, which puts the coefficients of the method in a table as follows: </p> c 1 a 11 a 12 … a 1 s c 2 a 21 a 22 … a 2 s ⋮ ⋮ ⋮ ⋱ ⋮ c s a s 1 a s 2 … a s s b 1 b 2 … b s {\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}}} <p>For <a href="/facts/Runge%E2%80%93Kutta_methods/YRjbyHpY">adaptive</a> and implicit methods, the Butcher tableau is extended to give values of b i ∗ {\displaystyle b_{i}^{*}} , and the estimated error is then </p> e n + 1 = h ∑ i = 1 s ( b i − b i ∗ ) k i {\displaystyle e_{n+1}=h\sum _{i=1}^{s}(b_{i}-b_{i}^{*})k_{i}} .