View source for List of Runge–Kutta methods

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

:<math>\frac{d y}{d t} = f(t, y).</math>

[[Explicit and implicit methods|Explicit]] Runge–Kutta methods take the form

:<math>\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 [[Explicit and implicit methods|implicit methods]] of s stages take the more general form, with the [[Explicit and implicit methods#Computation|solution to be found]] over all s

:<math>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>
\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 [[Runge–Kutta methods#Adaptive Runge–Kutta methods|adaptive]] and [[#Implicit_Runge–Kutta_methods|implicit methods]], the Butcher tableau is extended to give values of <math>b^*_i</math>, and the estimated error is then
:<math> 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>[a_{ij}]</math> is lower [[triangular matrix|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.

:<math>
\begin{array}{c|c}
0 & 0 \\
\hline
  & 1 \\
\end{array}
</math>

===Second-order methods===

====Generic second-order method====
Second-order methods can be generically written as follows:<ref name="butcher">{{cite book | last1=Butcher | first1=John C. | author1-link = John C. Butcher | title=Numerical Methods for Ordinary Differential Equations | publisher=John Wiley | isbn=978-0-471-96758-3 | year=2003}}</ref>

:<math>
\begin{array}{c|ccc}
0   & 0   & 0   \\
\alpha & \alpha & 0   \\
\hline
    & 1-\frac{1}{2\alpha} & \frac{1}{2\alpha} \\
\end{array}
</math>

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):

: <math>
\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:

: <math>
\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<ref name="ralston">{{cite journal |last1=Ralston |first1=Anthony |title=Runge-Kutta Methods with Minimum Error Bounds |journal=Math. Comput. |date=1962 |volume=16 |issue=80 |pages=431–437|doi=10.1090/S0025-5718-1962-0150954-0 |doi-access=free }}</ref> with two stages and a minimum local error bound:

: <math>
\begin{array}{c|cc}
0   & 0   & 0  \\
2/3 & 2/3 & 0  \\
\hline
    & 1/4   & 3/4  \\
\end{array}
</math>

===Third-order methods===

====Generic third-order method====
Third-order methods can be generically written as follows:<ref name="butcher"/>
 
: <math>
\begin{array}{c|ccc} 
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\\ 
\hline
 & 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}
</math>

with ''α'' ≠ 0, ''α'' ≠ {{Frac|2|3}}, ''β'' ≠ 0, and ''α'' ≠ ''β''.

====Kutta's third-order method====

: <math>
\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>

====Heun's third-order method====

:<math>
\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>

====Ralston's third-order method====

Ralston's third-order method<ref name="ralston"/> has a minimum local error bound and is used in the embedded [[Bogacki–Shampine method]].

:<math>
\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>

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

:<math>
\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>

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

:<math>
\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>

===Fourth-order methods===

====Classic fourth-order method====

The "original" Runge–Kutta method.<ref name="kutta">{{cite journal |last1=Kutta | first1=Martin | author1-link=Martin Kutta | title=Beitrag zur näherungsweisen Integration totaler Differentialgleichungen | journal=Zeitschrift für Mathematik und Physik | volume=46 | pages=435–453 | year=1901}}</ref>

:<math>
\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).<ref name="kutta" />

:<math>
\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<ref name="ralston"/> has minimum truncation error.
:<math>\begin{array}{c|cccc}
0   & 0   & 0   & 0   & 0\\
\frac{2}{5} & \frac{2}{5} & 0   & 0   & 0\\
\frac{14 - 3 \sqrt{5}}{16} & \frac{-2\,889 + 1\,428\sqrt{5}}{1\,024}   & \frac{3\,785 - 1\,620\sqrt{5}}{1\,024} & 0   & 0\\
1   & \frac{-3\,365 + 2\,094\sqrt{5}}{6\,040}   & \frac{-975 - 3\,046\sqrt{5}}{2\,552}   & \frac{467\,040 + 203\,968\sqrt{5}}{240\,845}   & 0\\
\hline
    & \frac{263 + 24\sqrt{5}}{1\,812} & \frac{125 - 1000\sqrt{5}}{3\,828} & \frac{3\,426\,304 + 1\,661\,952\sqrt{5}}{5\,924\,787} & \frac{30 - 4\sqrt{5}}{123}\\
\end{array}</math>

=== Fifith-order methods ===

==== Nyström's fifth-order method ====
This fifth-order method was a correction of the one proposed originally by Kutta's work.<ref>{{Cite journal |last=Butcher |first=J. C. |date=1996-03-01 |title=A history of Runge-Kutta methods |url=https://linkinghub.elsevier.com/retrieve/pii/0168927495001085 |journal=Applied Numerical Mathematics |volume=20 |issue=3 |pages=247–260 |doi=10.1016/0168-9274(95)00108-5 |issn=0168-9274}}</ref>

<math>
\begin{array}{c|cccccc}
0   & 0   & 0   & 0   & 0 & 0 & 0\\
\frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & 0 & 0\\
\frac{2}{5} & \frac{4}{25} & \frac{6}{25} & 0 & 0 & 0 & 0\\
1 & \frac{1}{4} & -3 & \frac{15}{4} & 0 & 0 & 0\\
\frac{2}{3} & \frac{2}{27} & \frac{10}{9} & -\frac{50}{81} & \frac{8}{81} & 0 & 0\\
\frac{4}{5} & \frac{2}{25} & \frac{12}{15} & \frac{2}{15} & \frac{8}{75} & 0 & 0\\
\hline
    & \frac{23}{192} & 0 & \frac{125}{192} & 0 & -\frac{27}{64} & \frac{125}{192}\\
\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>    y^*_{n+1} = y_n + h\sum_{i=1}^s b^*_i k_i, </math>

where the <math>k_i</math> are the same as for the higher order method. Then the error is

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

which is <math>O(h^p)</math>. The Butcher Tableau for this kind of method is extended to give the values of <math>b^*_i</math>
:<math>
\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>
\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|Fehlberg method]]<ref>{{Cite report
| last = Fehlberg
| first = E.
| date = July 1969
| title = Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems
| type = NASA Technical Report R-315
| url = https://ntrs.nasa.gov/search.jsp?R=19690021375
}}</ref> has two methods of orders 1 and 2. Its extended Butcher Tableau is:
{| cellpadding="3px" cellspacing="0px"
| width="20px" | || style="border-right:1px solid;" | 0
|- 
||| style="border-right:1px solid;" | 1/2 || 1/2
|- 
||| style="border-right:1px solid; border-bottom:1px solid;" |1|| style="border-bottom:1px solid;" |1/256|| style="border-bottom:1px solid;"|255/256 || style="border-bottom:1px solid;" |
|-
||| style="border-right:1px solid;" | ||1/512||255/256||1/512
|- 
||| style="border-right:1px solid;" | ||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:
{| cellpadding=3px cellspacing=0px
|width="20px"| || style="border-right:1px solid;" | 0
|- 
||| style="border-right:1px solid;" | 1/2 || 1/2
|- 
||| style="border-right:1px solid;" | 3/4 || 0 || 3/4
|- 
||| style="border-right:1px solid; border-bottom:1px solid;" | 1 || style="border-bottom:1px solid;" | 2/9 || style="border-bottom:1px solid;" | 1/3 || style="border-bottom:1px solid;" | 4/9 || style="border-bottom:1px solid;" | 
|- 
||| style="border-right:1px solid;" | || 2/9 || 1/3 || 4/9 || 0
|-
||| style="border-right:1px solid;" | || 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>\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

{| cellpadding=3px cellspacing=0px
|width="20px"| || style="border-right:1px solid;" | 0
|- 
||| style="border-right:1px solid;" | 1/5 || 1/5
|- 
||| style="border-right:1px solid;" | 3/10 || 3/40 || 9/40
|- 
||| style="border-right:1px solid;" | 3/5 || 3/10 || −9/10 || 6/5
|- 
||| style="border-right:1px solid;" | 1 || −11/54 || 5/2 || −70/27 || 35/27
|- 
||| style="border-right:1px solid; border-bottom:1px solid;" | 7/8 || style="border-bottom:1px solid;" | 1631/55296 || style="border-bottom:1px solid;" | 175/512 || style="border-bottom:1px solid;" | 575/13824 || style="border-bottom:1px solid;" | 44275/110592 || style="border-bottom:1px solid;" | 253/4096 || style="border-bottom:1px solid;" | 
|- 
||| style="border-right:1px solid;" | || 37/378 || 0 || 250/621 || 125/594 || 0 || 512/1771
|-
||| style="border-right:1px solid;" | || 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

{| cellpadding=3px cellspacing=0px
|width="20px"| || style="border-right:1px solid;" | 0
|- 
||| style="border-right:1px solid;" | 1/5 || 1/5
|- 
||| style="border-right:1px solid;" | 3/10 || 3/40 || 9/40
|- 
||| style="border-right:1px solid;" | 4/5  || 44/45 || −56/15 || 32/9
|- 
||| style="border-right:1px solid;" | 8/9  || 19372/6561 || −25360/2187 || 64448/6561 || −212/729
|- 
||| style="border-right:1px solid;" | 1 || 9017/3168 || −355/33 || 46732/5247 || 49/176 || −5103/18656
|-
||| style="border-right:1px solid; border-bottom:1px solid;" | 1 || style="border-bottom:1px solid;" | 35/384 || style="border-bottom:1px solid;" | 0 || style="border-bottom:1px solid;" | 500/1113 || style="border-bottom:1px solid;" | 125/192 || style="border-bottom:1px solid;" | −2187/6784 || style="border-bottom:1px solid;" | 11/84 || style="border-bottom:1px solid;" | 
|-
||| style="border-right:1px solid;" | || 35/384 || 0 || 500/1113 || 125/192 || −2187/6784 || 11/84 || 0
|- 
||| style="border-right:1px solid;" | || 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>
\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 method|Gauss-Legendre methods]]. It is a [[symplectic integrator]].

: <math>
\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>
\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>
\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>
\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;
<ref>For discussion see: {{cite journal |author1=Christopher A. Kennedy|author2=Mark H. Carpenter|title=Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review |journal=Technical Memorandum, NASA STI Program |date=2016|volume=|issue=|pages=|doi= |doi-access=|url=https://ntrs.nasa.gov/citations/20160005923}}</ref>
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>
\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>
\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>
\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 display="inline">x \ge \frac{1}{4}</math>.  Moreover, this method is [[L-stability|L-stable]] if and only if <math>x</math> equals one of the roots of the polynomial <math display="inline">x^2 - 2x + \frac{1}{2}</math>, i.e. if <math display="inline">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>x = 1/4</math>.

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

:<math>
\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 display="inline">x \ge \frac{1}{4}</math>.  As the previous method, this method is again L-stable if and only if <math>x</math> equals one of the roots of the polynomial <math display="inline">x^2 - 2x + \frac{1}{2}</math>, i.e. if <math display="inline">x = 1 \pm \frac{\sqrt2}{2}</math>. This condition is also necessary for 2nd order accuracy.

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

:<math>
\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>
\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 display="inline">\alpha = \frac{2}{\sqrt3}\cos{\frac{\pi}{18}}</math>.

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

:<math>
\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>x = 0.4358665215</math>

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

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

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

:<math>
\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,<ref name="Jay"/> 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]]<ref name="Jay"/> as a reference to the [[Gaussian quadrature#Gauss–Lobatto rules|Lobatto quadrature rule]], but were introduced by Byron L. Ehle in his thesis.<ref>{{harvtxt|Ehle|1969}}</ref>  All are implicit methods, have order 2''s''&nbsp;&minus;&nbsp;2 and they all have ''c''<sub>1</sub>&nbsp;=&nbsp;0 and ''c''<sub>''s''</sub>&nbsp;=&nbsp;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 method]]s. The second-order method is known as the [[trapezoidal rule (differential equations)|trapezoidal rule]]:

:<math>
\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>
\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 neither L-stable nor B-stable.<ref name="Jay" />

====Lobatto IIIB methods====

The Lobatto IIIB methods are not collocation methods, but they can be viewed as [[discontinuous collocation method]]s {{harv|Hairer|Lubich|Wanner|2006|loc=§II.1.4}}. The second-order method is given by

:<math>
\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>
\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 neither L-stable nor B-stable.<ref name="Jay" />

====Lobatto IIIC methods====

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

:<math>
\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>
\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.<ref name="Jay">See Laurent O. Jay (N.D.). [http://homepage.math.uiowa.edu/~ljay/publications.dir/Lobatto.pdf "Lobatto methods"]. [[University of Iowa]]</ref> The second-order method is given by

:<math>
\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>
\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>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> (\alpha_{A},\alpha_{B},\alpha_{C}) </math> by considering Lobatto coefficients of the form
:<math>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>\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>
\begin{array}{c|cc}
0   & 1/2 & 1/2\\
1   & -1/2 & 1/2 \\
\hline
    & 1/2 & 1/2 \\
\end{array}
</math>

and

:<math>
\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>\alpha_{A} = 2</math>, <math>\alpha_{B} = 2</math>, <math>\alpha_{C} = -1</math>, and <math>\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 2''s''&nbsp;&minus;&nbsp;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
:<math>
\begin{array}{c|cc}
0   & 1    \\
\hline
    & 1 \\
\end{array}
</math>

The third-order method is given by

:<math>
\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>
\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''<sub>i</sub> of this method are zeros of
:<math>\frac{d^{s-1}}{dx^{s-1}}(x^{s-1}(x-1)^s)</math>.

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

The third-order method is given by

:<math>
\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>
\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>

==Notes==
{{Reflist}}

==References==
* {{cite thesis|last1=Ehle|first1=Byron L.|title=On Padé approximations to the exponential function and A-stable methods for the numerical solution of initial value problems|url=https://cs.uwaterloo.ca/research/tr/1969/CS-RR-2010.pdf|date=1969}}
* {{Citation | last1=Hairer | first1=Ernst | last2=Nørsett | first2=Syvert Paul | last3=Wanner | first3=Gerhard | title=Solving ordinary differential equations I: Nonstiff problems | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-56670-0 | year=1993}}.
* {{Citation | last1=Hairer | first1=Ernst | last2=Wanner | first2=Gerhard | title=Solving ordinary differential equations II: Stiff and differential-algebraic problems | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-60452-5 | year=1996}}.
* {{Citation | last1=Hairer | first1=Ernst | last2=Lubich | first2=Christian | last3=Wanner | first3=Gerhard | title=Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-3-540-30663-4 | year=2006}}.

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