Source code for gala.integrate.pyintegrators.leapfrog

""" Leapfrog integration. """

# Third-party
import numpy as np

# Project
from ..core import Integrator
from ..timespec import parse_time_specification

__all__ = ["LeapfrogIntegrator"]
[docs]class LeapfrogIntegrator(Integrator): r""" A symplectic, Leapfrog integrator. Given a function for computing time derivatives of the phase-space coordinates, this object computes the orbit at specified times. .. seealso:: - http://en.wikipedia.org/wiki/Leapfrog_integration - http://ursa.as.arizona.edu/~rad/phys305/ODE_III/node11.html Naming convention for variables:: im1 = i-1 im1_2 = i-1/2 ip1 = i+1 ip1_2 = i+1/2 Examples -------- Using ``q`` as our coordinate variable and ``p`` as the conjugate momentum, we want to numerically solve for an orbit in the potential (Hamiltonian) .. math:: \Phi &= \frac{1}{2}q^2\\ H(q,p) &= \frac{1}{2}(p^2 + q^2) In this system, .. math:: \dot{q} &= \frac{\partial \Phi}{\partial p} = p \\ \dot{p} &= -\frac{\partial \Phi}{\partial q} = -q We will use the variable ``w`` to represent the full phase-space vector, :math:`w = (q,p)`. We define a function that computes the time derivates at any given time, ``t``, and phase-space position, ``w``:: def F(t,w): dw = [w[1], -w[0]] return dw .. note:: The force here is not time dependent, but this function always has to accept the independent variable (e.g., time) as the first argument. To create an integrator object, just pass this acceleration function in to the constructor, and then we can integrate orbits from a given vector of initial conditions:: integrator = LeapfrogIntegrator(acceleration) times,ws = integrator.run(w0=[1.,0.], dt=0.1, n_steps=1000) .. note:: When integrating a single vector of initial conditions, the return array will have 2 axes. In the above example, the returned array will have shape ``(2,1001)``. If an array of initial conditions are passed in, the return array will have 3 axes, where the last axis is for the individual orbits. Parameters ---------- func : func A callable object that computes the phase-space time derivatives at a time and point in phase space. func_args : tuple (optional) Any extra arguments for the derivative function. func_units : `~gala.units.UnitSystem` (optional) If using units, this is the unit system assumed by the integrand function. """
[docs] def step(self, t, x_im1, v_im1_2, dt): """ Step forward the positions and velocities by the given timestep. Parameters ---------- dt : numeric The timestep to move forward. """ x_i = x_im1 + v_im1_2 * dt F_i = self.F(t, np.vstack((x_i,v_im1_2)), *self._func_args) a_i = F_i[self.ndim:] v_i = v_im1_2 + a_i * dt / 2 v_ip1_2 = v_i + a_i * dt / 2 return x_i, v_i, v_ip1_2
def _init_v(self, t, w0, dt): """ Leapfrog updates the velocities offset a half-step from the position updates. If we're given initial conditions aligned in time, e.g. the positions and velocities at the same 0th step, then we have to initially scoot the velocities forward by a half step to prime the integrator. Parameters ---------- dt : numeric The first timestep. """ # here is where we scoot the velocity at t=t1 to v(t+1/2) F0 = self.F(t.copy(), w0.copy(), *self._func_args) a0 = F0[self.ndim:] v_1_2 = w0[self.ndim:] + a0*dt/2. return v_1_2
[docs] def run(self, w0, mmap=None, **time_spec): # generate the array of times times = parse_time_specification(self._func_units, **time_spec) n_steps = len(times) - 1 _dt = times[1] - times[0] w0_obj, w0, ws = self._prepare_ws(w0, mmap, n_steps) x0 = w0[:self.ndim] v0 = w0[self.ndim:] if _dt < 0.: v0 *= -1. dt = np.abs(_dt) else: dt = _dt # prime the integrator so velocity is offset from coordinate by a # half timestep v_im1_2 = self._init_v(times[0], w0, dt) x_im1 = x0 ws[:,0] = w0 for ii in range(1,n_steps+1): x_i, v_i, v_ip1_2 = self.step(times[ii], x_im1, v_im1_2, dt) ws[:self.ndim,ii,:] = x_i ws[self.ndim:,ii,:] = v_i x_im1, v_im1_2 = x_i, v_ip1_2 if _dt < 0: ws[self.ndim:,...] *= -1. return self._handle_output(w0_obj, times, ws)