This page summarizes numerical codes and research software that I make publicly available.

spectral-noisy-coupled-burgers-1d

spectral-noisy-coupled-burgers-1d is a C++ solver for one-dimensional multi-component noisy coupled Burgers equations.

The code solves conservative stochastic partial differential equations of the form

\[\partial_t \phi^\alpha + \sum_\beta A_{\alpha\beta}\partial_x\phi^\beta - \sum_\beta D_{\alpha\beta}\partial_x^2\phi^\beta + \sum_{\beta,\gamma}H^\alpha_{\beta\gamma}\partial_x(\phi^\beta\phi^\gamma) = \sum_\beta B_{\alpha\beta}\partial_x\xi^\beta .\]

Spatial derivatives are evaluated with a Fourier spectral method, and nonlinear terms are computed by a pseudospectral method with 3/2 padding. The solver supports conservative noise, nonlinear mode coupling, restart output, and several observables designed for fluctuating hydrodynamics and KPZ universality.

A distinctive feature of the code is that simulations are controlled by a LAMMPS-like input.script. This makes it easy to run different one-dimensional multi-component fluctuating hydrodynamic equations within the same framework. The Fourier spectral discretization also enables high-accuracy evaluation of spatial derivatives.

In particular, the code includes measurements of height fluctuation moments, skewness, excess kurtosis, Fourier-mode correlations, and time correlations. These observables are useful for testing KPZ/NFH scaling predictions directly from numerical simulations, and the code is designed so that new observables can be added without rewriting the solver core.

The examples demonstrate, among others,

  • comparison with finite-size exact results for the fluctuating advection-diffusion equation,
  • demonstration that height fluctuations in the one-component noisy Burgers equation approach the GOE Tracy-Widom-type distribution of the KPZ universality class, following settings such as Oliveira, Ferreira, and Alves, Phys. Rev. E 85, 010601 (2012),
  • reproduction of recent work on how height fluctuations in a two-component coupled Burgers equation are related to exact results for the KPZ equation, following De Nardis, Gopalakrishnan, and Vasseur, Phys. Rev. Lett. 131, 197102 (2023).

The repository is intended not only as a time-evolution solver, but also as a reproducible numerical framework: it includes input scripts, analysis scripts, and representative figures for comparing simulations with theoretical predictions.

Links

spectral-scalar-fluctuating-hydrodynamics-1d

I am planning to release another numerical code for more complex fluctuating hydrodynamic equations, restricted to one-component fields.

The target equations include conservative stochastic partial differential equations for a one-component scalar density field $\rho(x,t)$, such as

\[\frac{\partial \rho(x,t)}{\partial t} = \frac{\partial}{\partial x} \left[ D(\rho)\frac{\partial \rho}{\partial x} \right] - \frac{\partial}{\partial x} \left[ M(\rho)\frac{\partial^3 \rho}{\partial x^3} \right] - \frac{\partial}{\partial x} \left[ v(x)A(\rho) \right] + \frac{\partial}{\partial x} \left[ \sqrt{2\sigma(\rho)}\,\xi(x,t) \right].\]