# ode-rk4 [![Build Status](https://travis-ci.org/scijs/ode-rk4.svg)](https://travis-ci.org/scijs/ode-rk4) [![npm version](https://badge.fury.io/js/ode-rk4.svg)](http://badge.fury.io/js/ode-rk4) [![Dependency Status](https://david-dm.org/scijs/ode-rk4.svg)](https://david-dm.org/scijs/ode-rk4) > Integrate a system of ODEs using the Fourth Order Runge-Kutta (RK-4) method ## Introduction This module integrates a system of ordinary differential equations of the form

$\begin{eqnarray*} y'(t) &=& f(t, y(t)), \\ y(t_0) &=& y_0 \end{eqnarray*}$

where

is a vector of length

. Given time step $\Delta t$ , the Runge-Kutta 4 method integrates the ODE with update

$\begin{eqnarray*} y_{n+1} &=& \frac{\Delta t}{6}\left(k_1 + 2k_2 + 2k_3 + k_4\right) \\ t_{n+1} &=& t_n + \Delta t \end{eqnarray*}$

where

are given by

$\begin{eqnarray*} k_1 &=& f(t_n, y_n), \\ k_2 &=& f(t_n + \frac{\Delta t}{2}, y_n + \frac{\Delta t}{2} k_1), \\ k_3 &=& f(t_n + \frac{\Delta t}{2}, y_n + \frac{\Delta t}{2} k_2), \\ k_4 &=& f(t_n + \Delta t, y_n + \Delta tk_3). \end{eqnarray*}$

For a similar adaptive method using the fifth order Cash-Karp Runge-Kutta method with fourth order embedded error estimator, see [https://github.com/scijs/ode45-cash-karp](ode45-cash-karp]. ## Install ```bash $ npm install ode-rk4 ``` ## Example ```javascript var rk4 = require('ode-rk4') var deriv = function(dydt, y, t) { dydt[0] = -y[1] dydt[1] = y[0] } var y0 = [1,0] var n = 1000 var t0 = 0 var dt = 2.0 * Math.PI / n var integrator = rk4( y0, deriv, t0, dt ) // Integrate 1000 steps: integrator.steps(n) // Integrate all the way around a circle: // => integrator.y = [ 0.9999999999995743, -8.160481752145232e-11 ] ``` ## API ### `require('ode-rk4')( y0, deriv, t0, dt )` **Arguments:** - `y0`: an array or typed array containing initial conditions. This vector is updated in-place with each integrator step. - `deriv`: a function that calculates the derivative. Format is `function( dydt, y, t )`. Inputs are current state `y` and current time `t`, output is the calculated derivative `dydt`. - `t0`: initial time

. - `dt`: time step $\Delta t$ . **Returns**: Initialized integrator object. **Properties:** - `n`: dimension of `y0`. - `y`: current state. Initialized as a shallow copy of input `y0`. - `deriv`: function that calculates the derivative. Initialized from input. May be changed. - `t`: current time, incremented by `dt` with each time step. - `dt`: time step $\Delta t$ . Initialized from input `dt`. May be changed. **Methods:** - `.step()`: takes a single step of the RK-4 integrator and stores the result in-place in the `y` property. - `.steps( n )`: takes `n` steps of the RK-4 integrator, storing the result in-place in the `y` property. ## Credits (c) 2015 Ricky Reusser. MIT License