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scijs/ode-rk4

Name: ode-rk4

Owner: scijs

Description: Integrate a system of ODEs using the Fourth Order Runge-Kutta (RK-4) method

Created: 2015-07-24 16:13:52.0

Updated: 2018-05-22 20:49:01.0

Pushed: 2015-08-10 02:47:02.0

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Size: 272

Language: JavaScript

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README

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

m install ode-rk4

Example

rk4 = require('ode-rk4')

deriv = function(dydt, y, t) {
dt[0] = -y[1]
dt[1] =  y[0]


y0 = [1,0]
n = 1000
t0 = 0
dt = 2.0 * Math.PI / n

integrator = rk4( y0, deriv, t0, dt )

ntegrate 1000 steps:
grator.steps(n)

ntegrate 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

This work is supported by the National Institutes of Health's National Center for Advancing Translational Sciences, Grant Number U24TR002306. This work is solely the responsibility of the creators and does not necessarily represent the official views of the National Institutes of Health.