Numerical study of the Kardar-Parisi-Zhang equation

Abstract

We integrate numerically the Kardar-Parisi-Zhang (KPZ) equation in 1+1 and 2+1 dimensions using an Euler discretization scheme and the replacement of ${(\nabla h)}^2$ by exponentially decreasing functions of that quantity to suppress instabilities. When applied to the equation in 1+1 dimensions, the method of instability control provides values of scaling amplitudes consistent with exactly known results, in contrast to the deviations generated by the original scheme. In 2+1 dimensions, we spanned a range of the model parameters where transients with Edwards-Wilkinson or random growth are not bserved, in box sizes $8\le L\le 128$. We obtain roughness exponent $0.37\le \alpha \le 0.40$ and steady state height distributions with skewness $S=0.25\pm 0.01$ and kurtosis $Q=0.15\pm 0.1$. These estimates are obtained after extrapolations to the large $L$ limit, which is necessary due to significant finite-size effects in the estimates of effective exponents and height distributions. On the other hand, the steady state roughness distributions show weak scaling corrections and evidence of stretched exponentials tails. These results confirm previous estimates from lattice models, showing their reliability as representatives of the KPZ class.

arXiv comment17 pages, 6 figures, 1 table

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• Invited review: KPZ. Recent developments via a variational formulation
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• Vortex dynamics in a compact Kardar-Parisi-Zhang system
• Numerical Study of the Thermodynamic Uncertainty Relation for the KPZ-Equation
• Anomalous Roughening of Curvature-Driven Growth With a Variable Interface Window
• Steady-state skewness and kurtosis from renormalized cumulants in $(2+1)$-dimensional stochastic surface growth
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• Numerical Integration of the KPZ and Related Equations on Networks: The Case of the Cayley Tree
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