Numerical methods for ordinary differential equations

Definition

Let $f \colon [t_0, T] \times \R^d \to \R^d$ be continuous and Lipschitz continuous in its second argument. Then the initial value problem $y' = f(t, y)$, $y(t_0) = y_0$ has a unique solution $y$ on $[t_0, T]$. For an integer $N \ge 1$ let $h = (T - t_0)/N$ and $t_n = t_0 + nh$ for $n = 0, \ldots, N$. A $k$-step method is a rule that starts from given values $y_0, \ldots, y_{k-1}$. For $0 \le n \le N - k$ it determines $y_{n+k}$ from $h$, $t_n$, the values $y_n, \ldots, y_{n+k-1}$ and finitely many evaluations of $f$. It may do so by solving an equation that contains $y_{n+k}$. It is one-step if $k = 1$, explicit if $y_{n+k}$ is given by a formula in the earlier values, and implicit otherwise. The method is convergent if, for every such $f$ and starting values with $y_i \to y(t_0)$ for $i < k$ as $h \to 0$, $$\max_{0 \le n \le N} \norm{y(t_n) - y_n} \to 0 \quad \text{as } h \to 0.$$ It is convergent of order $p$ if this maximum is $O(h^p)$ for every sufficiently smooth $f$ and starting values with error $O(h^p)$.