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  "body": "\n## Definition\n\nLet $f \\colon [t_0, T] \\times \\R^d \\to \\R^d$ be continuous and Lipschitz continuous in its second argument.\nThen the [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$ has a unique solution $y$ on $[t_0, T]$.\nFor an integer $N \\ge 1$ let $h = (T - t_0)/N$ and $t_n = t_0 + nh$ for $n = 0, \\ldots, N$.\nA *$k$-step method* is a rule that starts from given values $y_0, \\ldots, y_{k-1}$.\nFor $0 \\le n \\le N - k$ it determines $y_{n+k}$ from $h$, $t_n$, the values $y_n, \\ldots, y_{n+k-1}$ and finitely many evaluations of $f$.\nIt may do so by solving an equation that contains $y_{n+k}$.\nIt is *one-step* if $k = 1$, *explicit* if $y_{n+k}$ is given by a formula in the earlier values, and *implicit* otherwise.\nThe method is *convergent* if, for every such $f$ and starting values with $y_i \\to y(t_0)$ for $i \u003c k$ as $h \\to 0$,\n$$\\max_{0 \\le n \\le N} \\norm{y(t_n) - y_n} \\to 0 \\quad \\text{as } h \\to 0.$$\nIt is convergent *of order* $p$ if this maximum is $O(h^p)$ for every sufficiently smooth $f$ and starting values with error $O(h^p)$.\n\n## Remarks\n\nExample: the [[Euler method|0JARTXX4XP]] $y_{n+1} = y_n + h f(t_n, y_n)$ is an explicit one-step method, convergent of order $1$ when $f$ is Lipschitz in $y$.\nExample: the two-step Adams–Bashforth method $y_{n+2} = y_{n+1} + \\frac{h}{2}\\big(3 f(t_{n+1}, y_{n+1}) - f(t_n, y_n)\\big)$ is explicit with $k = 2$.\nNon-example of convergence: the explicit two-step rule $y_{n+2} = -4 y_{n+1} + 5 y_n + h\\big(4 f(t_{n+1}, y_{n+1}) + 2 f(t_n, y_n)\\big)$ has local truncation error $O(h^4)$ but is not convergent, because the root $-5$ of $\\zeta^2 + 4\\zeta - 5$ makes perturbations grow like $5^n$.\nMany methods use variable steps $h_n = t_{n+1} - t_n$; then $h$ is replaced by $\\max_n h_n$.\n\n## Explanations\n\n### L1 human\n\nA numerical method for an initial value problem produces a list of approximate values on a grid of times, each computed from the previous ones.\nIt is convergent if the approximations get uniformly close to the true solution when the steps get small.\n",
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