List of Runge–Kutta methods

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

$\frac{d y}{d t} = f(t, y)\,$

which take the form

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

${\displaystyle k_{1}=f(t_{n},y_{n}),}$

${\displaystyle k_{2}=f(t_{n}+c_{2}h,y_{n}+h(a_{21}k_{1})),}$

${\displaystyle k_{3}=f(t_{n}+c_{3}h,y_{n}+h(a_{31}k_{1}+a_{32}k_{2})),}$

$\vdots$

${\displaystyle k_{i}=f\left(t_{n}+c_{i}h,y_{n}+h\sum _{j=1}^{i-1}a_{ij}k_{j}\right),}$

The methods listed on this page are each defined by its Butcher tableau, which puts the coefficients of the method in a table as follows:

$\begin{array}{c|cccc} c_1 & a_{11} & a_{12}& \dots & a_{1s}\\ c_2 & a_{21} & a_{22}& \dots & a_{2s}\\ \vdots & \vdots & \vdots& \ddots& \vdots\\ c_s & a_{s1} & a_{s2}& \dots & a_{ss} \\ \hline & b_1 & b_2 & \dots & b_s\\ \end{array}$

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 (also known as explicit trapezoid rule):

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

Ralston's method

Ralston's method is a second-order method 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

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

Kutta's third-order method

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

Classic fourth-order method

The "original" Runge–Kutta method.

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

3/8-rule fourth-order method

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

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

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

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

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

Bogacki–Shampine

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

	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

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

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 methods. It is a symplectic integrator.

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

Gauss–Legendre methods

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

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

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

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

Lobatto methods

There are three main families of Lobatto methods, called IIIA, IIIB and IIIC (in classial mathematical literature, the symbols I and II are reserved for two types of Radau methods). These are named after Rehuel Lobatto. 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:

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

The fourth-order method is given by

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

This 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

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

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

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

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.^[2] The second-order method is given by

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

The 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

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

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

$P_{s}(2x-1) - P_{s-1}(2x-1) = 0,$

where $P_{s}$ is the Legendre polynomial of degree 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

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

References

1. Fehlberg, E. (1969-07-01). "Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems".
2. http://homepage.math.uiowa.edu/~ljay/publications.dir/Lobatto.pdf

• 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.