Efficient Pseudo-spectral Algorithms for Statistical Field Theories

TL;DR

The paper introduces stochastic exponential time differencing (SETD) schemes for stiff SDEs and integrates them with a pseudospectral framework to efficiently simulate stochastic field theories with additive noise. By deriving SETD variants (EM, Milstein, SETD1, SETD2) and analyzing their convergence and stability, the authors provide robust, explicit methods that outperform traditional schemes in stiffness-laden Fourier-space simulations. They demonstrate the approach on Model A and B, KPZ, and Complex Ginzburg-Landau, detailing observables such as equal-time and dynamic correlators and susceptibility, and validate results against analytical predictions. Open-source code accompanies the work, facilitating high-precision, scalable studies of equilibrium and nonequilibrium SFTs.

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.

Figures (10)

• Figure 1: Steady state probability distribution of the position of the SAO for various numerical methods with varying ${h}$. All methods agree with the correct Boltzmann probability distribution for ${h}=0.01$. For ${h}=0.2$, EM method fails badly, whereas SETD-EM and SETD1 schemes fair much better. While SETD-EM results are closer to the analytical prediction for ${h}=0.2$, since it only approximates the variance up to $\mathcal{O}({h}^{2})$, we can observe some discrepancies, whereas SETD1 is free from such discrepancies. Parameters : $\Gamma=5$, $T=6$, $b=0.01$, $u_{0}=0$ averaged over an ensemble of size $S=10000$.
• Figure 2: Comparison of various algorithms for GBM. In (a,b), we show a single trajectory for different methods for an identical realisation of Wiener process. In (c), we compares the RMS error $\Delta(t)$ (see \ref{['eq:RMS-error']} for definition) for various schemes. Individual realizations for various algorithms can deviate from the analytical trajectory, but SETD schemes have significantly smaller RMS error compared to the EM and the Milstein method. Parameters : $\lambda=2$, $\mu=1$, $u_{0}=1$, $h=2^{-4}$, $\Delta(t)$ is averaged over an ensemble of size $S=1024$.
• Figure 3: Strong and weak convergence of various schemes for GBM \ref{['eq:gbm']} at $t=1$. Parameters : $\lambda=2$, $\mu=1$, $u_{0}=1$, $S=50000$.
• Figure 5: Stability of various schemes for GBM \ref{['eq:gbm']} in the $(\lambda h, \mu^2 h)$ plane, indicated by the shaded areas. For SETD schemes, we define $c=z\lambda$, $z \in [0,1]$, so that $\delta=(1-z)\lambda$. For $z=0$, the schemes derived in this work show the same stability as their non-ETD counterparts, shown in the leftmost plot, while for any $z > 0$ they have a larger region of stability.
• Figure 6: The correlation function $C_{\mathrm{ET}}(k)$, calculated as a time average in the steady state for (a) model A (total time $T = 10^5$) and (b) model B (total time $T = 10^6$) for different values of $r > 0$ in 1D. Parameters : $u = 0$, $N = 256$, $L = 64$, $D = 0.1$ and ${h} = 0.01$.