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

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

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

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.

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

Second-order methods can be generically written as follows:^[1]

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

with α ≠ 0.

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

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

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:

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

Ralston's method is a second-order method^[2] with two stages and a minimum local error bound:

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

Third-order methods can be generically written as follows:^[1]

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

with α ≠ 0, α ≠ 2⁄3, β ≠ 0, and α ≠ β.

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

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

Ralston's third-order method^[2] has a minimum local error bound and is used in the embedded Bogacki–Shampine method.

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

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

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

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

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

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]

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

This fourth order method^[2] has minimum truncation error.

${\displaystyle {\begin{array}{c|cccc}0&0&0&0&0\\{\frac {2}{5}}&{\frac {2}{5}}&0&0&0\\{\frac {14-3{\sqrt {5}}}{16}}&{\frac {-2\,889+1\,428{\sqrt {5}}}{1\,024}}&{\frac {3\,785-1\,620{\sqrt {5}}}{1\,024}}&0&0\\1&{\frac {-3\,365+2\,094{\sqrt {5}}}{6\,040}}&{\frac {-975-3\,046{\sqrt {5}}}{2\,552}}&{\frac {467\,040+203\,968{\sqrt {5}}}{240\,845}}&0\\\hline &{\frac {263+24{\sqrt {5}}}{1\,812}}&{\frac {125-1000{\sqrt {5}}}{3\,828}}&{\frac {3\,426\,304+1\,661\,952{\sqrt {5}}}{5\,924\,787}}&{\frac {30-4{\sqrt {5}}}{123}}\\\end{array}}}$

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

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

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

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

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

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

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:

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

The error estimate is used to control the stepsize.

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.

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.

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

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

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 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.

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.

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

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

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.

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

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

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

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

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

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

${\displaystyle {\begin{array}{c|ccc}{\frac {1}{2}}-{\frac {\sqrt {15}}{10}}&{\frac {5}{36}}&{\frac {2}{9}}-{\frac {\sqrt {15}}{15}}&{\frac {5}{36}}-{\frac {\sqrt {15}}{30}}\\{\frac {1}{2}}&{\frac {5}{36}}+{\frac {\sqrt {15}}{24}}&{\frac {2}{9}}&{\frac {5}{36}}-{\frac {\sqrt {15}}{24}}\\{\frac {1}{2}}+{\frac {\sqrt {15}}{10}}&{\frac {5}{36}}+{\frac {\sqrt {15}}{30}}&{\frac {2}{9}}+{\frac {\sqrt {15}}{15}}&{\frac {5}{36}}\\\hline &{\frac {5}{18}}&{\frac {4}{9}}&{\frac {5}{18}}\\&-{\frac {5}{6}}&{\frac {8}{3}}&-{\frac {5}{6}}\end{array}}}$

Diagonally Implicit Runge–Kutta (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:

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

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

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

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

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

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 ${\displaystyle x}$ 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 ${\displaystyle x=1/4}$.

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

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

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 ${\displaystyle x}$ equals one of the roots of the polynomial ${\textstyle x^{2}-2x+{\frac {1}{2}}}$, i.e. if ${\textstyle x=1\pm {\frac {\sqrt {2}}{2}}}$. This condition is also necessary for 2nd order accuracy.

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

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

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

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

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

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

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

with ${\displaystyle x=0.4358665215}$

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

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

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

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

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

There are three main families of Lobatto methods,^[6] 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.

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

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

The fourth-order method is given by

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

These methods are A-stable, but neither L-stable nor B-stable.^[6]

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

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

The fourth-order method is given by

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

Lobatto IIIB methods are A-stable, but neither L-stable nor B-stable.^[6]

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

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

The fourth-order method is given by

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

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

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

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

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

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

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

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

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

where

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

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

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

and

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

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

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 method is similar to the backward Euler method and given by

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

The third-order method is given by

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

The fifth-order method is given by

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

The c_i of this method are zeros of

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

The first-order method is equivalent to the backward Euler method.

The third-order method is given by

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

The fifth-order method is given by

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

• 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.