Difference between revisions of "Integrators for time stepping"
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| − | == | + | 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 | We are solving an initial value problem, given as | ||
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$y_{n+1} = y_{n} + \Delta t f(t, y_n)$ | $y_{n+1} = y_{n} + \Delta t f(t, y_n)$ | ||
| − | = Explicit methods = | + | == Explicit (single step) methods == |
| + | |||
| + | A family of single step methods are exaplicit Runge-Kutta methods | ||
| + | |||
| + | It is given by | ||
| + | :<math> y_{n+1} = y_n + h \sum_{i=1}^s b_i k_i, </math> | ||
| + | where<ref>{{harvnb|Press|Teukolsky|Vetterling|Flannery|2007|p=907}}</ref> | ||
| + | :<math> | ||
| + | \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} | ||
| + | </math> | ||
| + | :''(Note: the above equations have different but equivalent definitions in different texts).''<ref name=notation /> | ||
| + | |||
| + | To specify a particular method, one needs to provide the integer ''s'' (the number of stages), and the coefficients ''a<sub>ij</sub>'' (for 1 ≤ ''j'' < ''i'' ≤ ''s''), ''b<sub>i</sub>'' (for ''i'' = 1, 2, ..., ''s'') and ''c<sub>i</sub>'' (for ''i'' = 2, 3, ..., ''s''). The matrix [''a<sub>ij</sub>''] is called the ''Runge–Kutta matrix'', while the ''b<sub>i</sub>'' and ''c<sub>i</sub>'' are known as the ''weights'' and the ''nodes''.<ref>{{harvnb|Iserles|1996|p=38}}</ref> These data are usually arranged in a mnemonic device, known as a ''Butcher tableau'' (after [[John C. Butcher]]): | ||
Revision as of 11:54, 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 \[ y_{n+1} = y_n + h \sum_{i=1}^s b_i k_i, \] where[1] \[ \begin{align} k_1 & = f(t_n, y_n), \\ k_2 & = f(t_n+c_2h, y_n+h(a_{21}k_1)), \\ k_3 & = f(t_n+c_3h, y_n+h(a_{31}k_1+a_{32}k_2)), \\ & \ \ \vdots \\ k_s & = f(t_n+c_sh, y_n+h(a_{s1}k_1+a_{s2}k_2+\cdots+a_{s,s-1}k_{s-1})). \end{align} \]
- (Note: the above equations have different but equivalent definitions in different texts).[2]
- ↑ Template:Harvnb
- ↑ Cite error: Invalid
<ref>tag; no text was provided for refs namednotation - ↑ Template:Harvnb
