Butcher tableau

Summary

A Butcher tableau is the table of coefficients that specifies a Runge–Kutta method. It lists the nodes $c_i$ in a column, the matrix $(a_{ij})$ beside them, and the weights $b_i$ in a bottom row. Its shape shows at a glance whether the method is explicit, with $(a_{ij})$ strictly lower triangular, or implicit. Order conditions, stability functions and the algebraic theory of the Butcher group are all expressed in terms of these coefficients.