, and the estimated error is then The explicit methods are those where the matrix
The lack of stability and accuracy limits its popularity mainly to use as a simple introductory example of a numeric solution method.
Second-order methods can be generically written as follows:[1] with α ≠ 0.
[3] This fourth order method[2] has minimum truncation error.
This fifth-order method was a correction of the one proposed originally by Kutta's work.
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.
The Butcher Tableau for this kind of method is extended to give the values of
Its extended Butcher Tableau is: The error estimate is used to control the stepsize.
Its extended Butcher Tableau is: The first row of b coefficients gives the second-order accurate solution, and the second row has order one.
Its extended Butcher Tableau is: The first row of b coefficients gives the third-order accurate solution, and the second row has order two.
Its extended Butcher Tableau is: 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 and Karp have modified Fehlberg's original idea.
The extended tableau for the Cash–Karp method is The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four.
The extended tableau for the Dormand–Prince method is The first row of b coefficients gives the fifth-order accurate solution, and the second row gives the fourth-order accurate solution.
Unconditionally stable and non-oscillatory for linear diffusion problems.
The Gauss–Legendre method of order four has Butcher tableau: The Gauss–Legendre method of order six has Butcher tableau: Diagonally Implicit Runge–Kutta (DIRK) formulae have been widely used for the numerical solution of stiff initial value problems; [6] the advantage of this approach is that here the solution may be found sequentially as opposed to simultaneously.
Kraaijevanger and Spijker's two-stage Diagonally Implicit Runge–Kutta method: Qin and Zhang's two-stage, 2nd order, symplectic 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
Qin and Zhang's Diagonally Implicit Runge–Kutta method corresponds to Pareschi and Russo's Diagonally Implicit Runge–Kutta method with
Crouzeix's two-stage, 3rd order Diagonally Implicit Runge–Kutta method: Crouzeix's three-stage, 4th order Diagonally Implicit Runge–Kutta method: with
Three-stage, 3rd order, L-stable Diagonally Implicit Runge–Kutta method: with
Nørsett's three-stage, 4th order Diagonally Implicit Runge–Kutta method has the following Butcher tableau: with
gives the best stability properties for initial value problems.
Four-stage, 3rd order, L-stable Diagonally Implicit Runge–Kutta method There are three main families of Lobatto methods,[7] 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[7] as a reference to the Lobatto quadrature rule, but were introduced by Byron L. Ehle in his thesis.
Lobatto lived before the classic fourth-order method was popularized by Runge and Kutta.
They are also algebraically stable and thus B-stable, that makes them suitable for stiff problems.
One can consider a very general family of methods with three real parameters
Radau methods attain order 2s − 1 for s stages.
Radau methods are A-stable, but expensive to implement.