The explicit methods are those where the matrix 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.
Explicit midpoint method[]
The (explicit) midpoint method is a second-order method with two stages (see also the implicit midpoint method below):
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"):
Ralston's method[]
Ralston's method is a second-order method[1] with two stages and a minimum local error bound:
This fourth order method[1] has minimum truncation error.
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).
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
where the are the same as for the higher order method. Then the error is
which is . The Butcher Tableau for this kind of method is extended to give the values of
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:
The error estimate is used to control the stepsize.
Fehlberg RK1(2)[]
The Fehlberg method[3] 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 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.
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.
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.
Crank-Nicolson method[]
The Crank–Nicolson method corresponds to the implicit trapezoidal rule and is a second-order accurate and A-stable method.
The Gauss–Legendre method of order six has Butcher tableau:
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 simplest method from this class is the order 2 implicit midpoint method.
Kraaijevanger and Spijker's two-stage Diagonally Implicit Runge–Kutta method:
Pareschi and Russo's two-stage 2nd order Diagonally Implicit Runge–Kutta method:
This Diagonally Implicit Runge–Kutta method is A-stable if and only if . Moreover, this method is L-stable if and only if equals one of the roots of the polynomial , i.e. if .
Qin and Zhang's Diagonally Implicit Runge–Kutta method corresponds to Pareschi and Russo's Diagonally Implicit Runge–Kutta method with .
Two-stage 2nd order Diagonally Implicit Runge–Kutta method:
Again, this Diagonally Implicit Runge–Kutta method is A-stable if and only if . As the previous method, this method is again L-stable if and only if equals one of the roots of the polynomial , i.e. if .
Crouzeix's two-stage, 3rd order Diagonally Implicit Runge–Kutta method:
Nørsett's three-stage, 4th order Diagonally Implicit Runge–Kutta method has the following Butcher tableau:
with one of the three roots of the cubic equation . The three roots of this cubic equation are approximately , , and . The root gives the best stability properties for initial value problems.
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. All are implicit methods, have order 2s − 2 and they all have c1 = 0 and cs = 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.
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 (Hairer, Lubich & Wanner 2006, §II.1.4). The second-order method is given by
The fourth-order method is given by
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
The fourth-order method is given by
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.[4] The second-order method is given by
Butcher's three-stage, fourth-order method is given by
These methods are not A-stable, B-stable or L-stable. The Lobatto IIIC* method for is sometimes called the explicit trapezoidal rule.
Generalized Lobatto methods[]
One can consider a very general family of methods with three real parameters by considering Lobatto coefficients of the form
,
where
.
For example, Lobatto IIID family introduced in (Nørsett and Wanner, 1981), also called Lobatto IIINW, are given by
and
These methods correspond to , , , and . 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.