Difference between revisions of "Integrators for time stepping"
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| − | $y_{n+1} = y_n + h \sum_{i=1}^s b_i k_i$ | + | |
| + | $y_{n+1} = y_n + h \displaystyle \sum_{i=1}^s b_i k_i$$ | ||
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where | where | ||
Revision as of 11:55, 10 November 2017
This page describes how to solve ordinary differential equations numerically with examples from our library.
Introduction and notation
We are solving an initial value problem, given as
$ \begin{align*} \dot{y}(t) &= f(t, y) \\ y(t_0) &= y_0 \end{align*} $
where $y$ is the unknown (possibly vector) function, $t_0$ is the start time, $f$ is the derivative (the functions we wish to integrate) and $y_0$ is the initial value of $y$. Numerically, we usually choose a time step $\Delta t$ and integrate the function up to a certain time $t_{\max}$. Times os subsequent time steps are denoted with $t_i$ and function values with $y_i$.
The simplest method is explicit Euler's method: $y_{n+1} = y_{n} + \Delta t f(t, y_n)$
Explicit (single step) methods
A family of single step methods are exaplicit Runge-Kutta methods
It is given by
$y_{n+1} = y_n + h \displaystyle \sum_{i=1}^s b_i k_i$$ where $ \begin{align*} k_1 & = f(t_n, y_n), \\ k_2 & = f(t_n+c_2h, y_n+h(a_{21}k_1)), \\ k_3 & = f(t_n+c_3h, y_n+h(a_{31}k_1+a_{32}k_2)), \\ & \ \ \vdots \\ k_s & = f(t_n+c_sh, y_n+h(a_{s1}k_1+a_{s2}k_2+\cdots+a_{s,s-1}k_{s-1})). \end{align*} $
To specify a particular method, one needs to provide the integer s (the number of stages), and the coefficients aij (for 1 ≤ j < i ≤ s), bi (for i = 1, 2, ..., s) and ci (for i = 2, 3, ..., s). The matrix [aij] is called the Runge–Kutta matrix, while the bi and ci are known as the weights and the nodes.[1] These data are usually arranged in a mnemonic device, known as a Butcher tableau (after John C. Butcher):