Improved finite-difference and pseudospectral schemes for the Kardar-Parisi-Zhang equation with long-range temporal correlations

TL;DR

This study uses pseudospectral and improved finite-difference schemes to simulate the KPZ equation with long-range temporal correlations, revealing how such correlations influence scaling properties and universality classes in kinetic roughening.

Contribution

It introduces and compares two numerical schemes for simulating the KPZ equation with correlated noise, demonstrating their effectiveness and impact on understanding universality classes.

Findings

Scaling properties are affected by long-range temporal correlations.

The two schemes produce consistent results across regimes.

Long-range correlations can change the universality class.

Abstract

To investigate universal behavior and effects of long-range temporal correlations in kinetic roughening, we perform extensive simulations on the Kardar-Parisi-Zhang (KPZ) equation with temporally correlated noise based on pseudospectral (PS) and one of the improved finite-difference (FD) schemes. We find that scaling properties are affected by long-range temporal correlations within the effective temporally correlated regions. Our results are consistent with each other using these two independent numerical schemes, three characteristic roughness exponents (global roughness exponent $\alpha$, local roughness exponent $\alpha_{loc}$, and spectral roughness exponent $\alpha_{s}$) are approximately equal within the small temporally correlated regime, and satisfy $\alpha_{loc} \approx \alpha<\alpha_{s}$ for the large temporally correlated regime, and the difference between $\alpha_{s}$ and $\alpha$ increases with increasing the temporal correlation exponent $\theta$. Our results also show that PS and the improved FD schemes could effectively suppress numerical instabilities in the temporally correlated KPZ growth equation. Furthermore, our investigations suggest that when the effects of long-range temporal correlation are present, the continuum and discrete growth systems do not belong to the same universality class with the same temporal correlation exponent.