List of Runge–Kutta methods

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

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

Explicit Runge–Kutta methods take the form

{\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

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:

{\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 $b_{i}^{*}$ , and the estimated error is then

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

.

Explicit methods

The explicit methods are those where the matrix $[a_{ij}]$ is lower triangular.

Forward Euler

The Euler method is first order. The lack of stability and accuracy limits its popularity mainly to use as a simple introductory example of a numeric solution method.

{\begin{array}{c|c}0&0\\\hline &1\\\end{array}}

Explicit midpoint method

The (explicit) midpoint method is a second-order method with two stages (see also the implicit midpoint method below):

{\begin{array}{c|cc}0&0&0\\1/2&1/2&0\\\hline &0&1\\\end{array}}

Heun's method

Heun's method is a second-order method with two stages. It is also known as the explicit trapezoid rule, improved Euler's method, or modified Euler's method. (Note: The "eu" is pronounced the same way as in "Euler", so "Heun" rhymes with "coin"):

{\begin{array}{c|cc}0&0&0\\1&1&0\\\hline &1/2&1/2\\\end{array}}

Ralston's method

Ralston's method is a second-order method^[1] with two stages and a minimum local error bound:

{\begin{array}{c|cc}0&0&0\\2/3&2/3&0\\\hline &1/4&3/4\\\end{array}}

Generic second-order method

{\begin{array}{c|ccc}0&0&0\\\alpha &\alpha &0\\\hline &1-{\frac {1}{2\alpha }}&{\frac {1}{2\alpha }}\\\end{array}}

Kutta's third-order method

{\begin{array}{c|ccc}0&0&0&0\\1/2&1/2&0&0\\1&-1&2&0\\\hline &1/6&2/3&1/6\\\end{array}}

Generic third-order method

See Sanderse and Veldman (2019).^[2]

for α ≠ 0, 2⁄3, 1:

{\begin{array}{c|ccc}0&0&0&0\\\alpha &\alpha &0&0\\1&1+{\frac {1-\alpha }{\alpha (3\alpha -2)}}&-{\frac {1-\alpha }{\alpha (3\alpha -2)}}&0\\\hline &{\frac {1}{2}}-{\frac {1}{6\alpha }}&{\frac {1}{6\alpha (1-\alpha )}}&{\frac {2-3\alpha }{6(1-\alpha )}}\\\end{array}}

Heun's third-order method

{\begin{array}{c|ccc}0&0&0&0\\1/3&1/3&0&0\\2/3&0&2/3&0\\\hline &1/4&0&3/4\\\end{array}}

Van der Houwen's/Wray third-order method

{\begin{array}{c|ccc}0&0&0&0\\8/15&8/15&0&0\\2/3&1/4&5/12&0\\\hline &1/4&0&3/4\\\end{array}}

Ralston's third-order method

Ralston's third-order method^[1] is used in the embedded Bogacki–Shampine method.

{\begin{array}{c|ccc}0&0&0&0\\1/2&1/2&0&0\\3/4&0&3/4&0\\\hline &2/9&1/3&4/9\\\end{array}}

Third-order Strong Stability Preserving Runge-Kutta (SSPRK3)

{\begin{array}{c|ccc}0&0&0&0\\1&1&0&0\\1/2&1/4&1/4&0\\\hline &1/6&1/6&2/3\\\end{array}}

Classic fourth-order method

The "original" Runge–Kutta method.^[3]

{\begin{array}{c|cccc}0&0&0&0&0\\1/2&1/2&0&0&0\\1/2&0&1/2&0&0\\1&0&0&1&0\\\hline &1/6&1/3&1/3&1/6\\\end{array}}

3/8-rule fourth-order method

This method doesn't have as much notoriety as the "classic" method, but is just as classic because it was proposed in the same paper (Kutta, 1901).^[3]

{\begin{array}{c|cccc}0&0&0&0&0\\1/3&1/3&0&0&0\\2/3&-1/3&1&0&0\\1&1&-1&1&0\\\hline &1/8&3/8&3/8&1/8\\\end{array}}

Ralston's fourth-order method

This fourth order method^[1] has minimum truncation error.

{\begin{array}{c|cccc}0&0&0&0&0\\.4&.4&0&0&0\\.45573725&.29697761&.15875964&0&0\\1&.21810040&-3.05096516&3.83286476&0\\\hline &.17476028&-.55148066&1.20553560&.17118478\\\end{array}}

Embedded methods

The embedded methods are designed to produce an estimate of the local truncation error of a single Runge–Kutta step, and as result, allow to control the error with adaptive stepsize. This is done by having two methods in the tableau, one with order p and one with order p-1.

The lower-order step is given by

y_{n+1}^{*}=y_{n}+h\sum _{i=1}^{s}b_{i}^{*}k_{i},

where the $k_{i}$ are the same as for the higher order method. Then the error is

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

which is $O(h^{p})$ . The Butcher Tableau for this kind of method is extended to give the values of $b_{i}^{*}$

{\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}\\&b_{1}^{*}&b_{2}^{*}&\dots &b_{s}^{*}\\\end{array}}

Heun–Euler

The simplest adaptive Runge–Kutta method involves combining Heun's method, which is order 2, with the Euler method, which is order 1. Its extended Butcher Tableau is:

{\begin{array}{c|cc}0&\\1&1\\\hline &1/2&1/2\\&1&0\end{array}}

The error estimate is used to control the stepsize.

Fehlberg RK1(2)

The Fehlberg method^[4] has two methods of orders 1 and 2. Its extended Butcher Tableau is:

0
1/2	1/2
1	1/256	255/256
	1/512	255/256	1/512
	1/256	255/256	0

The first row of b coefficients gives the second-order accurate solution, and the second row has order one.

Bogacki–Shampine

The Bogacki–Shampine method has two methods of orders 2 and 3. Its extended Butcher Tableau is:

0
1/2	1/2
3/4	0	3/4
1	2/9	1/3	4/9
	2/9	1/3	4/9	0
	7/24	1/4	1/3	1/8

The first row of b coefficients gives the third-order accurate solution, and the second row has order two.

Fehlberg

The Runge–Kutta–Fehlberg method has two methods of orders 5 and 4; it is sometimes dubbed RKF45 . Its extended Butcher Tableau is:

{\begin{array}{r|ccccc}0&&&&&\\1/4&1/4&&&\\3/8&3/32&9/32&&\\12/13&1932/2197&-7200/2197&7296/2197&\\1&439/216&-8&3680/513&-845/4104&\\1/2&-8/27&2&-3544/2565&1859/4104&-11/40\\\hline &16/135&0&6656/12825&28561/56430&-9/50&2/55\\&25/216&0&1408/2565&2197/4104&-1/5&0\end{array}}

The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four. The coefficients here allow for an adaptive stepsize to be determined automatically.

Cash-Karp

Cash and Karp have modified Fehlberg's original idea. The extended tableau for the Cash–Karp method is

0
1/5	1/5
3/10	3/40	9/40
3/5	3/10	−9/10	6/5
1	−11/54	5/2	−70/27	35/27
7/8	1631/55296	175/512	575/13824	44275/110592	253/4096
	37/378	0	250/621	125/594	0	512/1771
	2825/27648	0	18575/48384	13525/55296	277/14336	1/4

The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four.

Dormand–Prince

The extended tableau for the Dormand–Prince method is

0
1/5	1/5
3/10	3/40	9/40
4/5	44/45	−56/15	32/9
8/9	19372/6561	−25360/2187	64448/6561	−212/729
1	9017/3168	−355/33	46732/5247	49/176	−5103/18656
1	35/384	0	500/1113	125/192	−2187/6784	11/84
	35/384	0	500/1113	125/192	−2187/6784	11/84	0
	5179/57600	0	7571/16695	393/640	−92097/339200	187/2100	1/40

The first row of b coefficients gives the fifth-order accurate solution, and the second row gives the fourth-order accurate solution.

Implicit methods

Backward Euler

The backward Euler method is first order. Unconditionally stable and non-oscillatory for linear diffusion problems.

{\begin{array}{c|c}1&1\\\hline &1\\\end{array}}

Implicit midpoint

The implicit midpoint method is of second order. It is the simplest method in the class of collocation methods known as the Gauss-Legendre methods. It is a symplectic integrator.

{\begin{array}{c|c}1/2&1/2\\\hline &1\end{array}}

Crank-Nicolson method

The Crank–Nicolson method corresponds to the implicit trapezoidal rule and is a second-order accurate and A-stable method.

{\begin{array}{c|cc}0&0&0\\1&1/2&1/2\\\hline &1/2&1/2\\\end{array}}

Gauss–Legendre methods

These methods are based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method of order four has Butcher tableau:

{\begin{array}{c|cc}{\frac {1}{2}}-{\frac {\sqrt {3}}{6}}&{\frac {1}{4}}&{\frac {1}{4}}-{\frac {\sqrt {3}}{6}}\\{\frac {1}{2}}+{\frac {\sqrt {3}}{6}}&{\frac {1}{4}}+{\frac {\sqrt {3}}{6}}&{\frac {1}{4}}\\\hline &{\frac {1}{2}}&{\frac {1}{2}}\\&{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}&{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}\\\end{array}}

The Gauss–Legendre method of order six has Butcher tableau:

{\begin{array}{c|ccc}{\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 {1}{2}}+{\frac {\sqrt {15}}{10}}&{\frac {5}{36}}+{\frac {\sqrt {15}}{30}}&{\frac {2}{9}}+{\frac {\sqrt {15}}{15}}&{\frac {5}{36}}\\\hline &{\frac {5}{18}}&{\frac {4}{9}}&{\frac {5}{18}}\\&-{\frac {5}{6}}&{\frac {8}{3}}&-{\frac {5}{6}}\end{array}}

Diagonally Implicit Runge–Kutta methods

Diagonally Implicit Runge–Kutta (DIRK) formulae have been widely used for the numerical solution of stiff initial value problems; ^[5] the advantage of this approach is that here the solution may be found sequentially as opposed to simultaneously.

The simplest method from this class is the order 2 implicit midpoint method.

Kraaijevanger and Spijker's two-stage Diagonally Implicit Runge–Kutta method:

{\begin{array}{c|cc}1/2&1/2&0\\3/2&-1/2&2\\\hline &-1/2&3/2\\\end{array}}

Qin and Zhang's two-stage, 2nd order, symplectic Diagonally Implicit Runge–Kutta method:

{\begin{array}{c|cc}1/4&1/4&0\\3/4&1/2&1/4\\\hline &1/2&1/2\\\end{array}}

Pareschi and Russo's two-stage 2nd order Diagonally Implicit Runge–Kutta method:

{\begin{array}{c|cc}x&x&0\\1-x&1-2x&x\\\hline &{\frac {1}{2}}&{\frac {1}{2}}\\\end{array}}

This Diagonally Implicit Runge–Kutta method is A-stable if and only if ${\textstyle x\geq {\frac {1}{4}}}$ . Moreover, this method is L-stable if and only if $x$ equals one of the roots of the polynomial ${\textstyle x^{2}-2x+{\frac {1}{2}}}$ , i.e. if ${\textstyle x=1\pm {\frac {\sqrt {2}}{2}}}$ . Qin and Zhang's Diagonally Implicit Runge–Kutta method corresponds to Pareschi and Russo's Diagonally Implicit Runge–Kutta method with $x=1/4$ .

Two-stage 2nd order Diagonally Implicit Runge–Kutta method:

{\begin{array}{c|cc}x&x&0\\1&1-x&x\\\hline &1-x&x\\\end{array}}

Again, this Diagonally Implicit Runge–Kutta method is A-stable if and only if ${\textstyle x\geq {\frac {1}{4}}}$ . As the previous method, this method is again L-stable if and only if $x$ equals one of the roots of the polynomial ${\textstyle x^{2}-2x+{\frac {1}{2}}}$ , i.e. if ${\textstyle x=1\pm {\frac {\sqrt {2}}{2}}}$ .

Crouzeix's two-stage, 3rd order Diagonally Implicit Runge–Kutta method:

{\begin{array}{c|cc}{\frac {1}{2}}+{\frac {\sqrt {3}}{6}}&{\frac {1}{2}}+{\frac {\sqrt {3}}{6}}&0\\{\frac {1}{2}}-{\frac {\sqrt {3}}{6}}&-{\frac {\sqrt {3}}{3}}&{\frac {1}{2}}+{\frac {\sqrt {3}}{6}}\\\hline &{\frac {1}{2}}&{\frac {1}{2}}\\\end{array}}

Crouzeix's three-stage, 4th order Diagonally Implicit Runge–Kutta method:

{\begin{array}{c|ccc}{\frac {1+\alpha }{2}}&{\frac {1+\alpha }{2}}&0&0\\{\frac {1}{2}}&-{\frac {\alpha }{2}}&{\frac {1+\alpha }{2}}&0\\{\frac {1-\alpha }{2}}&1+\alpha &-(1+2\,\alpha )&{\frac {1+\alpha }{2}}\\\hline &{\frac {1}{6\alpha ^{2}}}&1-{\frac {1}{3\alpha ^{2}}}&{\frac {1}{6\alpha ^{2}}}\\\end{array}}

with ${\textstyle \alpha ={\frac {2}{\sqrt {3}}}\cos {\frac {\pi }{18}}}$ .

Three-stage, 3rd order, L-stable Diagonally Implicit Runge–Kutta method:

{\begin{array}{c|ccc}x&x&0&0\\{\frac {1+x}{2}}&{\frac {1-x}{2}}&x&0\\1&-3x^{2}/2+4x-1/4&3x^{2}/2-5x+5/4&x\\\hline &-3x^{2}/2+4x-1/4&3x^{2}/2-5x+5/4&x\\\end{array}}

with $x=0.4358665215$

Nørsett's three-stage, 4th order Diagonally Implicit Runge–Kutta method has the following Butcher tableau:

{\begin{array}{c|ccc}x&x&0&0\\1/2&1/2-x&x&0\\1-x&2x&1-4x&x\\\hline &{\frac {1}{6(1-2x)^{2}}}&{\frac {3(1-2x)^{2}-1}{3(1-2x)^{2}}}&{\frac {1}{6(1-2x)^{2}}}\\\end{array}}

with $x$ one of the three roots of the cubic equation $x^{3}-3x^{2}/2+x/2-1/24=0$ . The three roots of this cubic equation are approximately $x_{1}=1.06858$ , $x_{2}=0.30254$ , and $x_{3}=0.12889$ . The root $x_{1}$ gives the best stability properties for initial value problems.

Four-stage, 3rd order, L-stable Diagonally Implicit Runge–Kutta method

{\begin{array}{c|cccc}1/2&1/2&0&0&0\\2/3&1/6&1/2&0&0\\1/2&-1/2&1/2&1/2&0\\1&3/2&-3/2&1/2&1/2\\\hline &3/2&-3/2&1/2&1/2\\\end{array}}

Lobatto methods

There are three main families of Lobatto methods, called IIIA, IIIB and IIIC (in classical mathematical literature, the symbols I and II are reserved for two types of Radau methods). These are named after Rehuel Lobatto^[6] as a reference to the Lobatto quadrature rule, but were introduced by Byron L. Ehle in his thesis.^[7] All are implicit methods, have order 2s − 2 and they all have c₁ = 0 and c_s = 1. Unlike any explicit method, it's possible for these methods to have the order greater than the number of stages. Lobatto lived before the classic fourth-order method was popularized by Runge and Kutta.

Lobatto IIIA methods

The Lobatto IIIA methods are collocation methods. The second-order method is known as the trapezoidal rule:

{\begin{array}{c|cc}0&0&0\\1&1/2&1/2\\\hline &1/2&1/2\\&1&0\\\end{array}}

The fourth-order method is given by

{\begin{array}{c|ccc}0&0&0&0\\1/2&5/24&1/3&-1/24\\1&1/6&2/3&1/6\\\hline &1/6&2/3&1/6\\&-{\frac {1}{2}}&2&-{\frac {1}{2}}\\\end{array}}

These methods are A-stable, but not L-stable and B-stable.

Lobatto IIIB methods

The Lobatto IIIB methods are not collocation methods, but they can be viewed as discontinuous collocation methods (Hairer, Lubich & Wanner 2006, §II.1.4). The second-order method is given by

{\begin{array}{c|cc}0&1/2&0\\1&1/2&0\\\hline &1/2&1/2\\&1&0\\\end{array}}

The fourth-order method is given by

{\begin{array}{c|ccc}0&1/6&-1/6&0\\1/2&1/6&1/3&0\\1&1/6&5/6&0\\\hline &1/6&2/3&1/6\\&-{\frac {1}{2}}&2&-{\frac {1}{2}}\\\end{array}}

Lobatto IIIB methods are A-stable, but not L-stable and B-stable.

Lobatto IIIC methods

The Lobatto IIIC methods also are discontinuous collocation methods. The second-order method is given by

{\begin{array}{c|cc}0&1/2&-1/2\\1&1/2&1/2\\\hline &1/2&1/2\\&1&0\\\end{array}}

The fourth-order method is given by

{\begin{array}{c|ccc}0&1/6&-1/3&1/6\\1/2&1/6&5/12&-1/12\\1&1/6&2/3&1/6\\\hline &1/6&2/3&1/6\\&-{\frac {1}{2}}&2&-{\frac {1}{2}}\\\end{array}}

They are L-stable. They are also algebraically stable and thus B-stable, that makes them suitable for stiff problems.

Lobatto IIIC* methods

The Lobatto IIIC* methods are also known as Lobatto III methods (Butcher, 2008), Butcher's Lobatto methods (Hairer et al., 1993), and Lobatto IIIC methods (Sun, 2000) in the literature.^[6] The second-order method is given by

{\begin{array}{c|cc}0&0&0\\1&1&0\\\hline &1/2&1/2\\\end{array}}

Butcher's three-stage, fourth-order method is given by

{\begin{array}{c|ccc}0&0&0&0\\1/2&1/4&1/4&0\\1&0&1&0\\\hline &1/6&2/3&1/6\\\end{array}}

These methods are not A-stable, B-stable or L-stable. The Lobatto IIIC* method for $s=2$ is sometimes called the explicit trapezoidal rule.

Generalized Lobatto methods

One can consider a very general family of methods with three real parameters $(\alpha _{A},\alpha _{B},\alpha _{C})$ by considering Lobatto coefficients of the form

a_{i,j}(\alpha _{A},\alpha _{B},\alpha _{C})=\alpha _{A}a_{i,j}^{A}+\alpha _{B}a_{i,j}^{B}+\alpha _{C}a_{i,j}^{C}+\alpha _{C*}a_{i,j}^{C*}

,

where

\alpha _{C*}=1-\alpha _{A}-\alpha _{B}-\alpha _{C}

.

For example, Lobatto IIID family introduced in (Nørsett and Wanner, 1981), also called Lobatto IIINW, are given by

{\begin{array}{c|cc}0&1/2&1/2\\1&-1/2&1/2\\\hline &1/2&1/2\\\end{array}}

and

{\begin{array}{c|ccc}0&1/6&0&-1/6\\1/2&1/12&5/12&0\\1&1/2&1/3&1/6\\\hline &1/6&2/3&1/6\\\end{array}}

These methods correspond to $\alpha _{A}=2$ , $\alpha _{B}=2$ , $\alpha _{C}=-1$ , and $\alpha _{C*}=-2$ . The methods are L-stable. They are algebraically stable and thus B-stable.

Radau methods

Radau methods are fully implicit methods (matrix A of such methods can have any structure). Radau methods attain order 2s − 1 for s stages. Radau methods are A-stable, but expensive to implement. Also they can suffer from order reduction. The first order Radau method is similar to backward Euler method.

Radau IA methods

The third-order method is given by

{\begin{array}{c|cc}0&1/4&-1/4\\2/3&1/4&5/12\\\hline &1/4&3/4\\\end{array}}

The fifth-order method is given by

{\begin{array}{c|ccc}0&{\frac {1}{9}}&{\frac {-1-{\sqrt {6}}}{18}}&{\frac {-1+{\sqrt {6}}}{18}}\\{\frac {3}{5}}-{\frac {\sqrt {6}}{10}}&{\frac {1}{9}}&{\frac {11}{45}}+{\frac {7{\sqrt {6}}}{360}}&{\frac {11}{45}}-{\frac {43{\sqrt {6}}}{360}}\\{\frac {3}{5}}+{\frac {\sqrt {6}}{10}}&{\frac {1}{9}}&{\frac {11}{45}}+{\frac {43{\sqrt {6}}}{360}}&{\frac {11}{45}}-{\frac {7{\sqrt {6}}}{360}}\\\hline &{\frac {1}{9}}&{\frac {4}{9}}+{\frac {\sqrt {6}}{36}}&{\frac {4}{9}}-{\frac {\sqrt {6}}{36}}\\\end{array}}

Radau IIA methods

The c_i of this method are zeros of

{\frac {d^{s-1}}{dx^{s-1}}}(x^{s-1}(x-1)^{s})

.

The third-order method is given by

{\begin{array}{c|cc}1/3&5/12&-1/12\\1&3/4&1/4\\\hline &3/4&1/4\\\end{array}}

The fifth-order method is given by

{\begin{array}{c|ccc}{\frac {2}{5}}-{\frac {\sqrt {6}}{10}}&{\frac {11}{45}}-{\frac {7{\sqrt {6}}}{360}}&{\frac {37}{225}}-{\frac {169{\sqrt {6}}}{1800}}&-{\frac {2}{225}}+{\frac {\sqrt {6}}{75}}\\{\frac {2}{5}}+{\frac {\sqrt {6}}{10}}&{\frac {37}{225}}+{\frac {169{\sqrt {6}}}{1800}}&{\frac {11}{45}}+{\frac {7{\sqrt {6}}}{360}}&-{\frac {2}{225}}-{\frac {\sqrt {6}}{75}}\\1&{\frac {4}{9}}-{\frac {\sqrt {6}}{36}}&{\frac {4}{9}}+{\frac {\sqrt {6}}{36}}&{\frac {1}{9}}\\\hline &{\frac {4}{9}}-{\frac {\sqrt {6}}{36}}&{\frac {4}{9}}+{\frac {\sqrt {6}}{36}}&{\frac {1}{9}}\\\end{array}}

Notes

1 2 3 Ralston, Anthony (1962). "Runge-Kutta Methods with Minimum Error Bounds". Math. Comput. 16 (80): 431–437. doi:10.1090/S0025-5718-1962-0150954-0.
↑ Sanderse, Benjamin; Veldman, Arthur (2019). "Constraint-consistent Runge–Kutta methods for one-dimensional incompressible multiphase flow". J. Comput. Phys. 384: 170. arXiv:1809.06114. Bibcode:2019JCoPh.384..170S. doi:10.1016/j.jcp.2019.02.001. S2CID 73590909.
1 2 Kutta, Martin (1901). "Beitrag zur näherungsweisen Integration totaler Differentialgleichungen". Zeitschrift für Mathematik und Physik. 46: 435–453.
↑ Fehlberg, E. (1969-07-01). "Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems". {{cite journal}}: Cite journal requires |journal= (help)
↑ For discussion see: Christopher A. Kennedy; Mark H. Carpenter (2016). "Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review". Technical Memorandum, NASA STI Program.
1 2 See Laurent O. Jay (N.D.). "Lobatto methods". University of Iowa
↑ Ehle (1969)

References

Ehle, Byron L. (1969). On Padé approximations to the exponential function and A-stable methods for the numerical solution of initial value problems (PDF) (Thesis).
Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0.
Hairer, Ernst; Wanner, Gerhard (1996), Solving ordinary differential equations II: Stiff and differential-algebraic problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-60452-5.
Hairer, Ernst; Lubich, Christian; Wanner, Gerhard (2006), Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-30663-4.

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[ralston-1] 1 2 3 Ralston, Anthony (1962). "Runge-Kutta Methods with Minimum Error Bounds". Math. Comput. 16 (80): 431–437. doi:10.1090/S0025-5718-1962-0150954-0.

[2] Sanderse, Benjamin; Veldman, Arthur (2019). "Constraint-consistent Runge–Kutta methods for one-dimensional incompressible multiphase flow". J. Comput. Phys. 384: 170. arXiv:1809.06114. Bibcode:2019JCoPh.384..170S. doi:10.1016/j.jcp.2019.02.001. S2CID 73590909.

[kutta-3] 1 2 Kutta, Martin (1901). "Beitrag zur näherungsweisen Integration totaler Differentialgleichungen". Zeitschrift für Mathematik und Physik. 46: 435–453.

[4] Fehlberg, E. (1969-07-01). "Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems". {{cite journal}}: Cite journal requires |journal= (help)

[5] For discussion see: Christopher A. Kennedy; Mark H. Carpenter (2016). "Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review". Technical Memorandum, NASA STI Program.

[Jay-6] 1 2 See Laurent O. Jay (N.D.). "Lobatto methods". University of Iowa

[7] Ehle (1969)

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Numerical methods for integration
First-order methods	Euler method Backward Euler Semi-implicit Euler Exponential Euler
Second-order methods	Verlet integration Velocity Verlet Trapezoidal rule Beeman's algorithm Midpoint method Heun's method Newmark-beta method Leapfrog integration
Higher-order methods	Exponential integrator Runge–Kutta methods List of Runge–Kutta methods Linear multistep method General linear methods Backward differentiation formula Yoshida Gauss–Legendre method
Theory	Symplectic integrator