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

$\begin{array}{rl}{y}_{n+1}& =y_n+h\sum _{i=1}^s{b}_i{k}_i\\ k_1& =f(t_n,y_n),\\ k_2& =f(t_n+c_{2}h,y_n+h(a_{21}{k}_1)),\\ k_3& =f(t_n+c_{3}h,y_n+h(a_{31}{k}_1+a_{32}{k}_2)),\\ & ⋮\\ k_i& =f(t_n+c_{i}h,y_n+h\sum _{j=1}^{i-1}{a}_{ij}{k}_j).\end{array}$

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

$k_i=f(t_n+c_{i}h,y_n+h\sum _{j=1}^s{a}_{ij}{k}_j).$

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

$\begin{array}{ccccc}{c}_1& a_{11}& a_{12}& \dots & a_{1s}\\ c_2& a_{21}& a_{22}& \dots & a_{2s}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ c_s& a_{s1}& a_{s2}& \dots & a_{ss}\\ & b_1& b_2& \dots & b_s\\ \end{array}$

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.

First-order methods

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.

$\begin{array}{cc}0& 0\\ & 1\\ \end{array}$

Second-order methods

Generic second-order method

Second-order methods can be generically written as follows:

$\begin{array}{ccc}0& 0& 0\\ \alpha & \alpha & 0\\ & 1-\frac{1}{2\alpha }& \frac{1}{2\alpha }\\ \end{array}$

with α ≠ 0.

Explicit midpoint method

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

$\begin{array}{ccc}0& 0& 0\\ 1/2& 1/2& 0\\ & 0& 1\\ \end{array}$

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:

$\begin{array}{ccc}0& 0& 0\\ 1& 1& 0\\ & 1/2& 1/2\\ \end{array}$

Ralston's method

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

$\begin{array}{ccc}0& 0& 0\\ 2/3& 2/3& 0\\ & 1/4& 3/4\\ \end{array}$

Third-order methods

Generic third-order method

Third-order methods can be generically written as follows:

$\begin{array}{cccc}0& 0& 0& 0\\ \alpha & \alpha & 0& 0\\ \beta & \frac{\beta }{\alpha }\frac{\beta -3\alpha (1-\alpha )}{(3\alpha -2)}& -\frac{\beta }{\alpha }\frac{\beta -\alpha }{(3\alpha -2)}& 0\\ & 1-\frac{3\alpha +3\beta -2}{6\alpha \beta }& \frac{3\beta -2}{6\alpha (\beta -\alpha )}& \frac{2-3\alpha }{6\beta (\beta -\alpha )}\\ \end{array}$

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

Kutta's third-order method

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

Heun's third-order method

$\begin{array}{cccc}0& 0& 0& 0\\ 1/3& 1/3& 0& 0\\ 2/3& 0& 2/3& 0\\ & 1/4& 0& 3/4\\ \end{array}$

Ralston's third-order method

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

$\begin{array}{cccc}0& 0& 0& 0\\ 1/2& 1/2& 0& 0\\ 3/4& 0& 3/4& 0\\ & 2/9& 1/3& 4/9\\ \end{array}$

Van der Houwen's/Wray's third-order method

$\begin{array}{cccc}0& 0& 0& 0\\ 8/15& 8/15& 0& 0\\ 2/3& 1/4& 5/12& 0\\ & 1/4& 0& 3/4\\ \end{array}$

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

$\begin{array}{cccc}0& 0& 0& 0\\ 1& 1& 0& 0\\ 1/2& 1/4& 1/4& 0\\ & 1/6& 1/6& 2/3\\ \end{array}$

Fourth-order methods

Classic fourth-order method

The "original" Runge–Kutta method.

$\begin{array}{ccccc}0& 0& 0& 0& 0\\ 1/2& 1/2& 0& 0& 0\\ 1/2& 0& 1/2& 0& 0\\ 1& 0& 0& 1& 0\\ & 1/6& 1/3& 1/3& 1/6\\ \end{array}$

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).

$\begin{array}{ccccc}0& 0& 0& 0& 0\\ 1/3& 1/3& 0& 0& 0\\ 2/3& -1/3& 1& 0& 0\\ 1& 1& -1& 1& 0\\ & 1/8& 3/8& 3/8& 1/8\\ \end{array}$

Ralston's fourth-order method

This fourth order method has minimum truncation error.

$\begin{array}{ccccc}0& 0& 0& 0& 0\\ \frac{2}{5}& \frac{2}{5}& 0& 0& 0\\ \frac{14-3\sqrt{5}}{16}& \frac{-2889+1428\sqrt{5}}{1024}& \frac{3785-1620\sqrt{5}}{1024}& 0& 0\\ 1& \frac{-3365+2094\sqrt{5}}{6040}& \frac{-975-3046\sqrt{5}}{2552}& \frac{467040+203968\sqrt{5}}{240845}& 0\\ & \frac{263+24\sqrt{5}}{1812}& \frac{125-1000\sqrt{5}}{3828}& \frac{3426304+1661952\sqrt{5}}{5924787}& \frac{30-4\sqrt{5}}{123}\\ \end{array}$

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 $b_i^{*}$

$\begin{array}{ccccc}{c}_1& a_{11}& a_{12}& \dots & a_{1s}\\ c_2& a_{21}& a_{22}& \dots & a_{2s}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ c_s& a_{s1}& a_{s2}& \dots & a_{ss}\\ & b_1& b_2& \dots & b_s\\ & b_1^{*}& b_2^{*}& \dots & b_s^{*}\\ \end{array}$

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:

$\begin{array}{cc}0& \\ 1& 1\\ & 1/2& 1/2\\ & 1& 0\end{array}$

The error estimate is used to control the stepsize.

Fehlberg RK1(2)

The Fehlberg method 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:

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

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.

$\begin{array}{cc}1& 1\\ & 1\\ \end{array}$

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.

$\begin{array}{cc}1/2& 1/2\end{array}$

Crank-Nicolson method

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

$\begin{array}{ccc}0& 0& 0\\ 1& 1/2& 1/2\\ & 1/2& 1/2\\ \end{array}$

Gauss–Legendre methods

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

$\begin{array}{ccc}\frac{1}{2}-\frac{\sqrt{3}}{6}& \frac{1}{4}& \frac{1}{4}-\frac{\sqrt{3}}{6}\\ \frac{1}{2}+\frac{\sqrt{3}}{6}& \frac{1}{4}+\frac{\sqrt{3}}{6}& \frac{1}{4}\\ & \frac{1}{2}& \frac{1}{2}\\ & \frac{1}{2}+\frac{\sqrt{3}}{2}& \frac{1}{2}-\frac{\sqrt{3}}{2}\\ \end{array}$

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

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

Diagonally Implicit Runge–Kutta methods

Diagonally Implicit Runge–Kutta (DIRK) formulae have been widely used for the numerical solution of stiff initial value problems; 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:

$\begin{array}{ccc}1/2& 1/2& 0\\ 3/2& -1/2& 2\\ & -1/2& 3/2\\ \end{array}$

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

$\begin{array}{ccc}1/4& 1/4& 0\\ 3/4& 1/2& 1/4\\ & 1/2& 1/2\\ \end{array}$

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

$\begin{array}{ccc}x& x& 0\\ 1-x& 1-2x& x\\ & \frac{1}{2}& \frac{1}{2}\\ \end{array}$

This Diagonally Implicit Runge–Kutta method is A-stable if and only if $x\ge \frac{1}{4}$. Moreover, this method is L-stable if and only if $x$ equals one of the roots of the polynomial $x^2-2x+\frac{1}{2}$, i.e. if $x=1\pm \frac{\sqrt{2}}{2}$. 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:

$\begin{array}{ccc}x& x& 0\\ 1& 1-x& x\\ & 1-x& x\\ \end{array}$

Again, this Diagonally Implicit Runge–Kutta method is A-stable if and only if $x\ge \frac{1}{4}$. As the previous method, this method is again L-stable if and only if $x$ equals one of the roots of the polynomial $x^2-2x+\frac{1}{2}$, i.e. if $x=1\pm \frac{\sqrt{2}}{2}$. This condition is also necessary for 2nd order accuracy. Crouzeix's two-stage, 3rd order Diagonally Implicit Runge–Kutta method:

$\begin{array}{ccc}\frac{1}{2}+\frac{\sqrt{3}}{6}& \frac{1}{2}+\frac{\sqrt{3}}{6}& 0\\ \frac{1}{2}-\frac{\sqrt{3}}{6}& -\frac{\sqrt{3}}{3}& \frac{1}{2}+\frac{\sqrt{3}}{6}\\ & \frac{1}{2}& \frac{1}{2}\\ \end{array}$

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

$\begin{array}{cccc}\frac{1+\alpha }{2}& \frac{1+\alpha }{2}& 0& 0\\ \frac{1}{2}& -\frac{\alpha }{2}& \frac{1+\alpha }{2}& 0\\ \frac{1-\alpha }{2}& 1+\alpha & -(1+2\alpha )& \frac{1+\alpha }{2}\\ & \frac{1}{6{\alpha }^2}& 1-\frac{1}{3{\alpha }^2}& \frac{1}{6{\alpha }^2}\\ \end{array}$

with $\alpha =\frac{2}{\sqrt{3}}\cos \frac{\pi }{18}$. Three-stage, 3rd order, L-stable Diagonally Implicit Runge–Kutta method:

$\begin{array}{cccc}x& x& 0& 0\\ \frac{1+x}{2}& \frac{1-x}{2}& x& 0\\ 1& -3x^2/2+4x-1/4& 3x^2/2-5x+5/4& x\\ & -3x^2/2+4x-1/4& 3x^2/2-5x+5/4& x\\ \end{array}$

with $x=0.4358665215$ Nørsett's three-stage, 4th order Diagonally Implicit Runge–Kutta method has the following Butcher tableau:

$\begin{array}{cccc}x& x& 0& 0\\ 1/2& 1/2-x& x& 0\\ 1-x& 2x& 1-4x& x\\ & \frac{1}{6(1-2x{)}^2}& \frac{3(1-2x{)}^2-1}{3(1-2x{)}^2}& \frac{1}{6(1-2x{)}^2}\\ \end{array}$

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

$\begin{array}{ccccc}1/2& 1/2& 0& 0& 0\\ 2/3& 1/6& 1/2& 0& 0\\ 1/2& -1/2& 1/2& 1/2& 0\\ 1& 3/2& -3/2& 1/2& 1/2\\ & 3/2& -3/2& 1/2& 1/2\\ \end{array}$

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 as a reference to the Lobatto quadrature rule, but were introduced by Byron L. Ehle in his thesis. 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:

$\begin{array}{ccc}0& 0& 0\\ 1& 1/2& 1/2\\ & 1/2& 1/2\\ & 1& 0\\ \end{array}$

The fourth-order method is given by

$\begin{array}{cccc}0& 0& 0& 0\\ 1/2& 5/24& 1/3& -1/24\\ 1& 1/6& 2/3& 1/6\\ & 1/6& 2/3& 1/6\\ & -\frac{1}{2}& 2& -\frac{1}{2}\\ \end{array}$

These methods are A-stable, but neither L-stable nor B-stable

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

$\begin{array}{ccc}0& 1/2& 0\\ 1& 1/2& 0\\ & 1/2& 1/2\\ & 1& 0\\ \end{array}$

The fourth-order method is given by

$\begin{array}{cccc}0& 1/6& -1/6& 0\\ 1/2& 1/6& 1/3& 0\\ 1& 1/6& 5/6& 0\\ & 1/6& 2/3& 1/6\\ & -\frac{1}{2}& 2& -\frac{1}{2}\\ \end{array}$

Lobatto IIIB methods are A-stable, but neither L-stable nor B-stable.

Lobatto IIIC methods

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

$\begin{array}{ccc}0& 1/2& -1/2\\ 1& 1/2& 1/2\\ & 1/2& 1/2\\ & 1& 0\\ \end{array}$

The fourth-order method is given by

$\begin{array}{cccc}0& 1/6& -1/3& 1/6\\ 1/2& 1/6& 5/12& -1/12\\ 1& 1/6& 2/3& 1/6\\ & 1/6& 2/3& 1/6\\ & -\frac{1}{2}& 2& -\frac{1}{2}\\ \end{array}$

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. The second-order method is given by

$\begin{array}{ccc}0& 0& 0\\ 1& 1& 0\\ & 1/2& 1/2\\ \end{array}$

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

$\begin{array}{cccc}0& 0& 0& 0\\ 1/2& 1/4& 1/4& 0\\ 1& 0& 1& 0\\ & 1/6& 2/3& 1/6\\ \end{array}$

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

$\begin{array}{ccc}0& 1/2& 1/2\\ 1& -1/2& 1/2\\ & 1/2& 1/2\\ \end{array}$

and

$\begin{array}{cccc}0& 1/6& 0& -1/6\\ 1/2& 1/12& 5/12& 0\\ 1& 1/2& 1/3& 1/6\\ & 1/6& 2/3& 1/6\\ \end{array}$

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.

Radau IA methods

The first order method is similar to the backward Euler method and given by

$\begin{array}{cc}0& 1\\ & 1\\ \end{array}$

The third-order method is given by

$\begin{array}{ccc}0& 1/4& -1/4\\ 2/3& 1/4& 5/12\\ & 1/4& 3/4\\ \end{array}$

The fifth-order method is given by

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

Radau IIA methods

The c_i of this method are zeros of

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

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

$\begin{array}{ccc}1/3& 5/12& -1/12\\ 1& 3/4& 1/4\\ & 3/4& 1/4\\ \end{array}$

The fifth-order method is given by

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

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.