List of Runge–Kutta methods explained

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

(d y)/(d t) = f(t, y).

Explicit Runge–Kutta methods take the form

\begin{align} y _{n+1} & = y _{n} + h \sum ^{s} _{i =1} b _{i} k _{i} \\ k _{1} & = f(t _{n,} y _{n),} \\ k _{2} & = f(t _{n+c} _{2h,} y _{n+h(a} _{21} k _{1)),} \\ k _{3} & = f(t _{n+c} _{3h,} 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 ^{i -1} _{j =1} a _{ij} k _{j\right). \end{align}}

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 ^{s} _{j =1} 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} & ... & a _{1s} \\ c _{2} & a _{21} & a _{22} & ... & a _{2s} \\ \vdots & \vdots & \vdots& \ddots& \vdots\\ c _{s} & a _{s1} & a _{s2} & ... & a _{ss} \\ \hline & b _{1} & b _{2} & ... & 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 thene _{n+1} = h\sum ^{s} _{i =1} (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:

\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- (1)/(2 \alpha) & (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,, 1:

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

\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 \\ (2)/(5) & (2)/(5) & 0 & 0 & 0 \\ (14-3 \sqrt{ 5)/() } & \frac & \frac & 0 & 0\\1 & \frac & \frac & \frac & 0\\\hline & \frac & \frac & \frac & \frac\\\end

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 ^{s} _{i =1} b ^{*} _{i} k _{i,}

where the

k _{i} are the same as for the higher order method. Then the error ise _{n+1} = y _{n+1} - y ^{*} _{n +1} = h\sum ^{s} _{i =1} (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} & ... & a _{1s} \\ c _{2} & a _{21} & a _{22} & ... & a _{2s} \\ \vdots & \vdots & \vdots& \ddots& \vdots\\ c _{s} & a _{s1} & a _{s2} & ... & a _{ss} \\ \hline & b _{1} & b _{2} & ... & b _{s\\} & b ^{*} _{1} & b ^{*} _{2} & ... & b ^{*\\ \end{array}} _{s}

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} (1)/(- 2) (\sqrt 3)/(6) & (1)/(4) & (1)/(- 4) (\sqrt 3)/(\\ 6) (1)/(+ 2) (\sqrt 3)/(6) & (1)/(+ 4) (\sqrt 3)/(& 6) (1)/(4) \\ \hline & (1)/(2) & (1)/(2) \\ & (12+ (\sqrt 3)/(2))/(&) (12- (\sqrt 3)/(2))/(\\ \end{array})

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

\begin{array}{c|ccc} (1)/(2) - (\sqrt{ 15)/() } & \frac & \frac- \frac & \frac - \frac \\\frac & \frac + \frac & \frac & \frac - \frac\\\frac + \frac & \frac + \frac & \frac + \frac & \frac \\\hline & \frac & \frac & \frac \\ & -\frac56 & \frac83 & -\frac56\end

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 - 2 x & x \\ \hline & (1)/(2) & (1)/(2) \\ \end{array}

This Diagonally Implicit Runge–Kutta method is A-stable if and only if $x\; \backslash ge\; \backslash frac$. Moreover, this method is L-stable if and only if

x equals one of the roots of the polynomial $x^2\; -\; 2x\; +\; \backslash frac$, i.e. if $x\; =\; 1\; \backslash pm\; \backslash frac$.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 $x\; \backslash ge\; \backslash frac$. As the previous method, this method is again L-stable if and only if

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

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

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

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

\begin{array}{c|ccc} (1+ \alpha)/(2) & (1+ \alpha)/(2) & 0 & 0 \\ (1)/(2) & - (\alpha)/(2) & (1+ \alpha)/(2) & 0 \\ (1- \alpha)/(2) & 1+ \alpha & - ( 1+2 \alpha) & (1+ \alpha)/(2) \\\hline & (1)/(6 \alpha ^{2}) & 1 - (1)/(3 \alpha ^{2}) & (1)/(6 \alpha ^{2}) \\ \end{array} with $\backslash alpha\; =\; \backslash frac\backslash cos$.

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

\begin{array}{c|ccc} x & x & 0 & 0 \\ (1+ x)/(2) & (1- x)/(2) & x & 0 \\ 1 & -3 x ^{2/2+4x-1/4} & 3 x ^{2/2-5x+5/4} & x \\ \hline & -3 x ^{2/2+4x-1/4} & 3 x ^{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 & 2 x & 1-4 x & x \\ \hline & (1)/(6 ( 1-2 x) ^{2}) & (3 ( 1-2 x) ^{2} -1)/(3 ( 1-2 x) ^{2}) & (1)/(6 ( 1-2 x) ^{2}) \\ \end{array}

with

x one of the three roots of the cubic equation x ^{3} -3 x ^{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 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}{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 \\ & - (12)/(&) 2 & - (12)/(\\ \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 . 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 \\ & - (12)/(&) 2 & - (12)/(\\ \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 \\ & - (12)/(&) 2 & - (12)/(\\ \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 forma _{i,j} (\alpha _{A} ,\alpha _{B} ,\alpha _{C} ) = \alpha _{A} a ^{A} _{i,j} + \alpha _{B} a ^{B} _{i,j} + \alpha _{C} a ^{C} _{i,j} + \alpha _{C*} a ^{C*} _{i,j} ,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 & (1)/(9) & (-1- \sqrt{ 6)/() } & \frac \\\frac - \frac & \frac & \frac + \frac & \frac - \frac\\\frac + \frac & \frac & \frac + \frac & \frac - \frac \\\hline & \frac & \frac + \frac & \frac - \frac \\\end

Radau IIA methods

The c_i of this method are zeros of

(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} (2)/(5) - (\sqrt{ 6)/() } & \frac - \frac & \frac - \frac & -\frac + \frac \\\frac + \frac & \frac + \frac & \frac + \frac & -\frac - \frac\\1 & \frac - \frac & \frac + \frac & \frac \\\hline & \frac - \frac & \frac + \frac & \frac \\\end

References

• Ehle. Byron L.. On Padé approximations to the exponential function and A-stable methods for the numerical solution of initial value problems. 1969.
• .
• .
• .

Notes and References

1. Ralston . Anthony . Runge-Kutta Methods with Minimum Error Bounds . Math. Comput. . 1962 . 16 . 80 . 431–437. 10.1090/S0025-5718-1962-0150954-0 . free .
2. Sanderse . Benjamin . Veldman . Arthur . Constraint-consistent Runge–Kutta methods for one-dimensional incompressible multiphase flow . . 2019. 384 . 170 . 10.1016/j.jcp.2019.02.001 . 1809.06114 . 2019JCoPh.384..170S . 73590909 .
3. Kutta . Martin . Martin Kutta . Beitrag zur näherungsweisen Integration totaler Differentialgleichungen . Zeitschrift für Mathematik und Physik . 46 . 435–453 . 1901.
4. Fehlberg. E.. July 1969. Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems. NASA Technical Report R-315.
5. For discussion see: Christopher A. Kennedy. Mark H. Carpenter. Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review . Technical Memorandum, NASA STI Program . 2016.
6. See Laurent O. Jay (N.D.). "Lobatto methods". University of Iowa