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.

17 pages, 6 figures, 1 table

References in corpus (4)

• Pseudospectral versus finite-differences schemes in the numerical integration of stochastic models of surface growth
• Finite-size effects in roughness distribution scaling
• Aging dynamics of non-linear elastic interfaces: the Kardar-Parisi-Zhang equation
• Universality classes of the Kardar-Parisi-Zhang equation

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• Numerical Study of the Thermodynamic Uncertainty Relation for the KPZ-Equation
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• Steady-state skewness and kurtosis from renormalized cumulants in $(2+1)$-dimensional stochastic surface growth
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