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 k_n 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 ode45-cash-karp.

Install

$ npm install ode-rk4

Example

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

Repository ode-rk4

Ana Paula Diniz Marques