Pseudospectral versus finite-differences schemes in the numerical integration of stochastic models of surface growth

TL;DR

This paper compares pseudospectral and finite difference schemes for numerically integrating stochastic surface growth models, finding pseudospectral methods more stable and accurate, especially in avoiding numerical instabilities and multiscaling artifacts.

Contribution

It demonstrates that pseudospectral schemes outperform finite differences in stability and accuracy for stochastic surface growth models in 1+1 dimensions.

Findings

Pseudospectral method is more stable for both KPZ and LDV models.

Results from pseudospectral method align better with continuum predictions.

Finite difference methods exhibit instabilities leading to artificial multiscaling.

Abstract

We present a comparison between finite differences schemes and a pseudospectral method applied to the numerical integration of stochastic partial differential equations that model surface growth. We have studied, in 1+1 dimensions, the Kardar, Parisi and Zhang model (KPZ) and the Lai, Das Sarma and Villain model (LDV). The pseudospectral method appears to be the most stable for a given time step for both models. This means that the time up to which we can follow the temporal evolution of a given system is larger for the pseudospectral method. Moreover, for the KPZ model, a pseudospectral scheme gives results closer to the predictions of the continuum model than those obtained through finite difference methods. On the other hand, some numerical instabilities appearing with finite difference methods for the LDV model are absent when a pseudospectral integration is performed. These numerical instabilities give rise to an approximate multiscaling observed in the numerical simulations. With the pseudospectral approach no multiscaling is seen in agreement with the continuum model.