List of Runge–Kutta methods

Short description: none

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

[math]\displaystyle{ \frac{d y}{d t} = f(t, y). }[/math]

Explicit Runge–Kutta methods take the form

[math]\displaystyle{ \begin{align} 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_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_{j = 1}^{i-1} a_{ij} k_j\right). \end{align} }[/math]

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

[math]\displaystyle{ k_i = f\left(t_n + c_i h, y_n + h \sum_{j = 1}^{s} a_{ij} k_j\right). }[/math]

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

[math]\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} }[/math]

For adaptive and implicit methods, the Butcher tableau is extended to give values of [math]\displaystyle{ b^*_i }[/math], and the estimated error is then

[math]\displaystyle{ e_{n+1} = h\sum_{i=1}^s (b_i - b^*_i) k_i }[/math].

Explicit methods

The explicit methods are those where the matrix [math]\displaystyle{ [a_{ij}] }[/math] 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.

[math]\displaystyle{ \begin{array}{c|c} 0 & 0 \\ \hline & 1 \\ \end{array} }[/math]

Explicit midpoint method

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

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

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. (Note: The "eu" is pronounced the same way as in "Euler", so "Heun" rhymes with "coin"):

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

Ralston's method

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

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

Generic second-order method

[math]\displaystyle{ \begin{array}{c|ccc} 0 & 0 & 0 \\ \alpha & \alpha & 0 \\ \hline & 1-\frac{1}{2\alpha} & \frac{1}{2\alpha} \\ \end{array} }[/math]

Kutta's third-order method

[math]\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} }[/math]

Generic third-order method

See Sanderse and Veldman (2019).

for α ≠ 0, ​²⁄₃, 1:

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

Heun's third-order method

[math]\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} }[/math]

Van der Houwen's/Wray third-order method

[math]\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} }[/math]

Ralston's third-order method

Ralston's third-order method is used in the embedded Bogacki–Shampine method.

[math]\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} }[/math]

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

[math]\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} }[/math]

Classic fourth-order method

The "original" Runge–Kutta method.

[math]\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} }[/math]

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).

[math]\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} }[/math]

Ralston's fourth-order method

This fourth order method has minimum truncation error.

[math]\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} }[/math]

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

[math]\displaystyle{ y^*_{n+1} = y_n + h\sum_{i=1}^s b^*_i k_i, }[/math]

where the [math]\displaystyle{ k_i }[/math] are the same as for the higher order method. Then the error is

[math]\displaystyle{ e_{n+1} = y_{n+1} - y^*_{n+1} = h\sum_{i=1}^s (b_i - b^*_i) k_i, }[/math]

which is [math]\displaystyle{ O(h^p) }[/math]. The Butcher Tableau for this kind of method is extended to give the values of [math]\displaystyle{ b^*_i }[/math]

[math]\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} }[/math]

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:

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

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:

[math]\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} }[/math]

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.

[math]\displaystyle{ \begin{array}{c|c} 1 & 1 \\ \hline & 1 \\ \end{array} }[/math]

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.

[math]\displaystyle{ \begin{array}{c|c} 1/2 & 1/2 \\ \hline & 1 \end{array} }[/math]

Crank-Nicolson method

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

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

Gauss–Legendre methods

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

[math]\displaystyle{ \begin{array}{c|cc} \frac{1}{2}-\frac{\sqrt3}{6} & \frac{1}{4} & \frac{1}{4}-\frac{\sqrt3}{6} \\ \frac{1}{2}+\frac{\sqrt3}{6} & \frac{1}{4}+\frac{\sqrt3}{6} &\frac{1}{4} \\ \hline & \frac{1}{2} & \frac{1}{2}\\ & \frac12+\frac{\sqrt3}{2} & \frac12-\frac{\sqrt3}{2} \\ \end{array} }[/math]

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

[math]\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} \\ & -\frac56 & \frac83 & -\frac56 \end{array} }[/math]

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:

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

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

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

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

[math]\displaystyle{ \begin{array}{c|cc} x & x & 0 \\ 1 - x & 1 - 2x & x \\ \hline & \frac{1}{2} & \frac{1}{2}\\ \end{array} }[/math]

This Diagonally Implicit Runge–Kutta method is A-stable if and only if [math]\displaystyle{ x \ge \frac{1}{4} }[/math]. Moreover, this method is L-stable if and only if [math]\displaystyle{ x }[/math] equals one of the roots of the polynomial [math]\displaystyle{ x^2 - 2x + \frac{1}{2} }[/math], i.e. if [math]\displaystyle{ x = 1 \pm \frac{\sqrt2}{2} }[/math]. Qin and Zhang's Diagonally Implicit Runge–Kutta method corresponds to Pareschi and Russo's Diagonally Implicit Runge–Kutta method with [math]\displaystyle{ x = 1/4 }[/math].

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

[math]\displaystyle{ \begin{array}{c|cc} x & x & 0 \\ 1 & 1 - x & x \\ \hline & 1 - x & x\\ \end{array} }[/math]

Again, this Diagonally Implicit Runge–Kutta method is A-stable if and only if [math]\displaystyle{ x \ge \frac{1}{4} }[/math]. As the previous method, this method is again L-stable if and only if [math]\displaystyle{ x }[/math] equals one of the roots of the polynomial [math]\displaystyle{ x^2 - 2x + \frac{1}{2} }[/math], i.e. if [math]\displaystyle{ x = 1 \pm \frac{\sqrt2}{2} }[/math].

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

[math]\displaystyle{ \begin{array}{c|cc} \frac{1}{2}+\frac{\sqrt3}{6} & \frac{1}{2}+\frac{\sqrt3}{6} & 0 \\ \frac{1}{2}-\frac{\sqrt3}{6} & -\frac{\sqrt3}{3} & \frac{1}{2}+\frac{\sqrt3}{6} \\ \hline & \frac{1}{2} & \frac{1}{2}\\ \end{array} }[/math]

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

[math]\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} }[/math]

with [math]\displaystyle{ \alpha = \frac{2}{\sqrt3}\cos{\frac{\pi}{18}} }[/math].

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

[math]\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} }[/math]

with [math]\displaystyle{ x = 0.4358665215 }[/math]

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

[math]\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} }[/math]

with [math]\displaystyle{ x }[/math] one of the three roots of the cubic equation [math]\displaystyle{ x^3 -3x^2/2 + x/2 - 1/24 = 0 }[/math]. The three roots of this cubic equation are approximately [math]\displaystyle{ x_1 = 1.06858 }[/math], [math]\displaystyle{ x_2 = 0.30254 }[/math], and [math]\displaystyle{ x_3 = 0.12889 }[/math]. The root [math]\displaystyle{ x_1 }[/math] gives the best stability properties for initial value problems.

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

[math]\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} }[/math]

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:

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

The fourth-order method is given by

[math]\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 \\ & -\frac12 & 2 & -\frac12 \\ \end{array} }[/math]

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 (Hairer Lubich). The second-order method is given by

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

The fourth-order method is given by

[math]\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 \\ & -\frac12 & 2 & -\frac12 \\ \end{array} }[/math]

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

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

The fourth-order method is given by

[math]\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 \\ & -\frac12 & 2 & -\frac12 \\ \end{array} }[/math]

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

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

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

[math]\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} }[/math]

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

Generalized Lobatto methods

One can consider a very general family of methods with three real parameters [math]\displaystyle{ (\alpha_{A},\alpha_{B},\alpha_{C}) }[/math] by considering Lobatto coefficients of the form

[math]\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*} }[/math],

where

[math]\displaystyle{ \alpha_{C*} = 1 - \alpha_{A} - \alpha_{B} - \alpha_{C} }[/math].

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

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

and

[math]\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} }[/math]

These methods correspond to [math]\displaystyle{ \alpha_{A} = 2 }[/math], [math]\displaystyle{ \alpha_{B} = 2 }[/math], [math]\displaystyle{ \alpha_{C} = -1 }[/math], and [math]\displaystyle{ \alpha_{C*} = -2 }[/math]. 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

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

The fifth-order method is given by

[math]\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} }[/math]

Radau IIA methods

The c_i of this method are zeros of

[math]\displaystyle{ \frac{d^{s-1}}{dx^{s-1}}(x^{s-1}(x-1)^s) }[/math].

The third-order method is given by

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

The fifth-order method is given by

[math]\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} }[/math]

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 .