I have written some python code which was designed to try to solve the following differential equation: $ $ \ddot{x}+\omega_0^2x=\eta(t),$ $ where $ \eta(t)$ is the gaussian white noise, with mean 0 and variance 1. The initial conditions are: $ $ x(0)=\dot{x}(0)=0.$ $ The code is given here:
import numpy as np import matplotlib.pyplot as plt from scipy.stats import norm class HarmonicOdeSolver: def __init__(self, dt, x0, xd0, omega_squared): "Inits the solver." self.dt = dt self.dt_squared = dt ** 2 self.t = dt self.omega_squared = omega_squared self.x0 = x0 self.xd0 = xd0 self.x = [xd0 * dt + x0, x0] def step(self): "Steps the solver." xt, xtm1 = self.x xtp1 = (2 - self.omega_squared * self.dt_squared) * xt - xtm1 \ + self.dt_squared * norm.rvs() self.x = (xtp1, xt) self.t += self.dt def step_until(self, tmax, snapshot_dt): "Steps the solver until a given time, returns snapshots." ts = [self.t] vals = [self.x[0]] niter = max(1, int(snapshot_dt // self.dt)) while self.t < tmax: for _ in range(niter): self.step() vals.append(self.x[0]) ts.append(self.t) return np.array(ts), np.array(vals) solver = HarmonicOdeSolver(1e-2, 0, 0, 1) snapshot_dt = 1.0 ts, vals = solver.step_until(1000, snapshot_dt) plt.plot(ts, np.sqrt(vals ** 2)) plt.plot(ts, np.sqrt(ts / 2)) The code was taken from and explained here. I naively hoped that I could simply add the following line of code:
self.dt_squared * norm.rvs() to simulate Gaussian white noise. One problem I have noticed is that the results appear to be highly dependent on the time step used. In a similar post we found that the variance of the oscillator should grow as: $ $ \sqrt{\langle x(t)^2\rangle}\sim\sqrt{\frac{t}{2}}.$ $ I would like to reproduce this result, does anyone know of a simple way to simulate a harmonic oscillator driven by white noise?
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