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Tableau

The Tableau type is the main data structure in RungeKutta.jl. It holds all the coeffieints and information on a Runge-Kutta method in the form of a so-called Butcher tableau

\[\begin{array}{c|c} c & a \\ \hline & b^{T} \\ \end{array} = \begin{array}{c|cccc} c_{1} & a_{11} & a_{12} & \dots & a_{1s} \\ c_{2} & a_{21} & a_{22} & \dots & a_{2s} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ c_{s} & a_{s1} & a_{s2} & \dots & a_{ss} \\ \hline & b_{1} & b_{2} & \dots & b_{s} \\ \end{array}\]

The Tableau type has the following fields:

  • name is a descriptive name of the tableau,
  • o the order of the method,
  • s the number of stages,
  • a the coefficients,
  • b the weights,
  • c the nodes.

Although one of the main goals of RungeKutta.jl is to provide as many known tableaus as possible (see below), it is straight-forward to create a custom tableau. The tableau of Heun's method, for example, is defined as follows:

a = [[0.0  0.0]
     [1.0  0.0]]
b =  [0.5, 0.5]
c =  [0.0, 1.0]
o = 2

tab = Tableau(:heun, o, a, b, c)

There exist several constructors

Tableau{T}(name, o, s, a, b, c)
Tableau{T}(name, o, a, b, c)
Tableau(name,o, s, a, b, c)
Tableau(name,o, a, b, c)

The meaning of the arguments corresponds to the fields as explained above and T is the datatype of the coefficient arrays. If s is ommitted it is inferred from the length of c. All coefficient arrays are checked for compatible sizes. If T is not specified, the datatype is inferred from the type of a, b and c, which are then assumed to be identical. If T is specified and does not correspond to the types of a, b and c, the arrays are converted accordingly.