List of Runge–Kutta methods

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

:$frac\{d\; y\}\{d\; t\}\; =\; f(t,\; y),$

which take the form

:$y\_\{n+1\}\; =\; y\_n\; +\; h\; sum\_\{i=1\}^s\; b\_i\; k\_i,$

:$k\_i\; =\; fleft(t\_n\; +\; c\_i\; h,\; y\_n\; +\; h\; sum\_\{j\; =\; 1\}^s\; a\_\{ij\}\; k\_j\; ight).$

The methods listed on this page are each defined by its Butcher Tableau, which puts the coefficients of the method in a table as follows:

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Explicit methods

The explicit methods are those where the matrix $[a\_\{ij\}]$ is lower triangular.

Forward Euler

This method is first order. The lack of stability and accuracy makes this popular primarily as a simple first introduction to numeric solution.

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Kutta's third-order method

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Classic fourth-order method

The "original" Runge–Kutta method.

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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

:$y^*\_\{n+1\}\; =\; y\_n\; +\; hsum\_\{i=1\}^s\; b^*\_i\; k\_i,$

where the $k\_i$ are the same as for the higher order method. Then the error is

:$e\_\{n+1\}\; =\; y\_\{n+1\}\; -\; y^*\_\{n+1\}\; =\; hsum\_\{i=1\}^s\; (b\_i\; -\; b^*\_i)\; k\_i,$

which is "O"("h" "p"). The Butcher Tableau for this kind of method is extended to give the values of $b^*\_i$:

Heun-Euler

The simplest adaptive Runge-Kutta method involves combining the Heun 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.

Bogacki–Shampine

The Bogacki–Shampine method has two methods of orders 3 and 2. Its extended Butcher Tableau is:

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

The first row of "b" coefficients gives the fifth-order accurate solution, and the second row has order four.

Implicit methods

Backward Euler

This method is first order. Unconditionally stable and non-oscillatory for linear diffusion problems.

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Lobatto methods

There are three families of Lobatto methods, called IIIA, IIIB and IIIC. All are implicit methods, have order $2\; s\; -\; 2$ and they all have $c\_1=0$ 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 closely analogous to the Crank–Nicolson method.

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The fourth-order method is given by

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Lobatto IIIB methods

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

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The fourth-order method is given by

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Lobatto IIIC methods

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

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The fourth-order method is given by

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References

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