Efficient Pseudo-spectral Algorithms for Statistical Field Theories

TL;DR

This paper introduces efficient stochastic pseudo-spectral algorithms with improved stability and convergence for simulating complex stochastic field theories, enabling accurate computation of physical observables.

Contribution

It develops explicit exponential time differencing schemes combined with pseudo-spectral methods for stable, accurate simulation of stochastic field theories with additive noise.

Findings

Schemes achieve strong convergence up to order O(h).

Algorithms effectively simulate models like KPZ and Ginzburg-Landau.

Open source implementation provided for practical use.

Abstract

We present stochastic variants of the exponential time differencing schemes for stiff stochastic differential equations. We derive three explicit schemes that offer better stability compared to Euler-Maruyama and Milstein's method, and achieve strong convergence up to order O(h) in the time step h. We combine these schemes with a pseudo-spectral approach to outline efficient algorithms for simulating stochastic field theories with additive noise. To illustrate the effectiveness of this approach, we study several systems in and out of equilibrium, including Model A, Model B, the Kardar-Parisi-Zhang equation, and the Complex Ginzburg-Landau equation. We outline procedures for computing physical observables such as the critical exponents, correlation functions, and dynamic linear response, and provide our implementation as open source code.