List of Runge–Kutta methods – Wikipedia

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

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

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

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

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

${displaystyle e_{n+1}=hsum _{i=1}^{s}(b_{i}-b_{i}^{*})k_{i}}$

.

Explicit methods[edit]

The explicit methods are those where the matrix

${displaystyle [a_{ij}]}$

is lower triangular.

Forward Euler[edit]

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.

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

Explicit midpoint method[edit]

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[edit]

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”):

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

Ralston’s method[edit]

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

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

Generic second-order method[edit]

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

Kutta’s third-order method[edit]

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

Generic third-order method[edit]

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

for α ≠ 0, 2⁄3, 1:

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

Heun’s third-order method[edit]

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

Van der Houwen’s/Wray third-order method[edit]

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

Ralston’s third-order method[edit]

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

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

Third-order Strong Stability Preserving Runge-Kutta (SSPRK3)[edit]

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

Classic fourth-order method[edit]

The “original” Runge–Kutta method.^[3]

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

3/8-rule fourth-order method[edit]

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]

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

Ralston’s fourth-order method[edit]

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

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

Embedded methods[edit]

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

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

where the

${displaystyle k_{i}}$

are the same as for the higher order method. Then the error is

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

which is

${displaystyle O(h^{p})}$

. The Butcher Tableau for this kind of method is extended to give the values of

${displaystyle b_{i}^{*}}$

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

Heun–Euler[edit]

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:

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

The error estimate is used to control the stepsize.

Fehlberg RK1(2)[edit]

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[edit]

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[edit]

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

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

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[edit]

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[edit]

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[edit]

Backward Euler[edit]

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

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

Implicit midpoint[edit]

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.

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

Crank-Nicolson method[edit]

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

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

Gauss–Legendre methods[edit]

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 methods[edit]

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

${textstyle xgeq {frac {1}{4}}}$

. 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

${textstyle x=1pm {frac {sqrt {2}}{2}}}$

.
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

${textstyle xgeq {frac {1}{4}}}$

. 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

${textstyle x=1pm {frac {sqrt {2}}{2}}}$

.

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:

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

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

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

Lobatto methods[edit]

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.^[6] 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[edit]

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 not L-stable and B-stable.

Lobatto IIIB methods[edit]

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 not L-stable and B-stable.

Lobatto IIIC methods[edit]

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.

Lobatto IIIC* methods[edit]

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.^[7] 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.

Generalized Lobatto methods[edit]

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[edit]

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[edit]

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

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

Radau IIA methods[edit]

The c_i of this method are zeros of

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

.

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

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

References[edit]

• 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.