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  "body": "\n## Summary\n\nA Butcher tableau is the table of coefficients that specifies a [[Runge–Kutta method|7DBPJKYC6C]].\nIt lists the nodes $c_i$ in a column, the matrix $(a_{ij})$ beside them, and the weights $b_i$ in a bottom row.\nIts shape shows at a glance whether the method is explicit, with $(a_{ij})$ strictly lower triangular, or implicit.\nOrder conditions, stability functions and the algebraic theory of the [[Butcher group|6MH1DHMCPF]] are all expressed in terms of these coefficients.\n\n## Explanations\n\n### L1 human\n\nA Runge–Kutta method mixes several slope evaluations within each step.\nThe Butcher tableau is a compact recipe that says where in the step each slope is measured and how the slopes are combined.\nFor example, the tableau with nodes $0, 1$, the single entry $a_{21} = 1$, and weights $\\frac12, \\frac12$ is [[Heun's method|RJEK60G28F]].\nDifferent tables give different methods, and catalogues of methods are usually written as lists of tableaux.\n",
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