Example phase shift signal extracted from the simulation.
The approximately linear growth indicates a small propagation-speed mismatch.
This repository contains a small reproducible numerical experiment studying phase drift in long-time symplectic simulations of a traveling wave in the periodic Fermi-Pasta-Ulam-Tsingou (FPUT-beta) lattice.
The main question is why the numerical error appears to grow over long time intervals. In this experiment, two explanations are separated:
- genuine deformation of the waveform
- accumulated phase drift caused by a small propagation-speed mismatch
The numerical evidence suggests that most of the observed error growth in this setup is caused by phase drift rather than visible deformation of the waveform.
The main diagnostic is to compare the numerical displacement profile against the reference wave before and after optimal spatial alignment.
| Diagnostic | Observed behavior |
|---|---|
| Direct waveform error | Grows over long time because the wave drifts relative to the fixed reference |
| Alignment-based waveform error | Remains much smaller after correcting for translation |
| Phase shift | Grows approximately linearly in time |
| Drift estimate | Small, typically on the order of 1e-5 in the tested runs |
| Energy drift | Remains very small under Störmer-Verlet time integration |
The interpretation is that the dominant long-time error is a coherent phase error, not an immediate loss of waveform shape.
We study the periodic FPUT-beta lattice with bond potential
phi(r) = 0.5 r^2 + 0.25 r^4
and force
phi'(r) = r + r^3
The equations of motion are
x_ddot_i = phi'(x_{i+1} - x_i) - phi'(x_i - x_{i-1})
with periodic boundary conditions.
A traveling-wave-like profile is computed using
- Fourier spectral discretization
- Newton continuation
- a discrete traveling-wave residual equation
Parameters used in the main experiment:
L = 16
k = pi / 16
speed shift target = 0.02
The reference wave speed is
c_ref = c0 + speed_shift
where
c0 = sqrt(2*(1 - cos(k))) / k
The base wave profile is repeated several times to create the full lattice used for time integration.
The system is integrated using the velocity-Verlet, or Störmer-Verlet, symplectic method.
Parameters used across the repo include:
smoke-test dt = 0.01
time-step validation dt = 5e-3, 2.5e-3, 1.25e-3, 6.25e-4
simulation length about 1000 wave periods for validation runs
sampling rate = 4 samples per period
Several diagnostics are recorded during the simulation.
Direct waveform error
Relative L2 difference between the numerical displacement profile and the fixed reference traveling wave profile.
Alignment-based error
For each sampled displacement profile, the optimal spatial shift s(t) is computed by minimizing
|| u(x,t) - u0(x + s) ||_2
Sub-grid translations are implemented using FFT interpolation.
Phase drift estimate
The shift signal is
- unwrapped,
- corrected by subtracting the expected translation,
- fit using linear regression.
The fitted slope gives a drift-rate estimate in this alignment-shift convention. Its sign depends on the convention used for the spatial shift, so it should be interpreted as the measured coherent phase-drift rate rather than a separately derived wave speed.
Energy drift
Energy is computed as
H = sum(0.5 * v_i^2) + sum(phi(x_{i+1} - x_i))
and remains nearly conserved throughout the simulation.
src/
build_traveling_wave_frac.m
run_phase_drift_from_tw_frac.m
nsoli.m
experiments/
validate_time_step.m
validate_repeats.m
validate_newton_tolerance.m
generate_figures.m
tests/
smoke_test.m
figures/
generated plots
archive/
earlier exploratory scripts
The src directory contains reusable functions.
The experiments directory contains scripts that run the numerical tests.
The tests directory contains lightweight checks for path and interface problems.
The figures directory stores generated plots.
From the repository root, run the following commands in MATLAB:
addpath('src')
run('experiments/validate_time_step.m')
run('experiments/validate_repeats.m')
run('experiments/validate_newton_tolerance.m')
run('experiments/generate_figures.m')The validation scripts write .mat and .csv result files. The figure-generation script reuses saved results if they already exist.
The expected output figures are:
figures/direct_vs_aligned_error.png
figures/phase_shift_vs_time.png
figures/energy_drift.png
figures/drift_vs_dt.png
figures/drift_vs_repeats.png
figures/drift_vs_newton_tol.png
A small smoke test is included for checking that the main MATLAB functions still run together on a short problem:
run('tests/smoke_test.m')This test is meant to catch basic path or interface issues. It is not a replacement for the longer validation sweeps in experiments.
This repository documents a reproducible numerical experiment rather than a polished software package. The emphasis is on clear numerical diagnostics and interpretation of the results.
The results should be read as evidence for this controlled FPUT-beta traveling-wave setup, not as a general theorem for all nonlinear lattices, amplitudes, or numerical methods. The main point is that translation alignment separates phase drift from waveform deformation in a useful and measurable way.