Integrators for time stepping

Integrators for time stepping

We are solving an initial value problem, given as

$ \begin{align*} \dot{y}(t) &= f(t, y) \\ y(t_0) &= y_0 \end{align*} $

where $y$ is the unknown (possibly vector) function, $t_0$ is the start time, $f$ is the derivative (the functions we wish to integrate) and $y_0$ is the initial value of $y$. Numerically, we usually choose a time step $\Delta t$ and integrate the function up to a certain time $t_{\max}$. Times os subsequent time steps are denoted with $t_i$ and function values with $y_i$.

The simplest method is explicit Euler's method: $y_{n+1} = y_{n} + \Delta t f(t, y_n)$