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

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  • Hu, Xiongpeng
  • Hao, Dapeng
  • Xia, Hui

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 α, local roughness exponent αloc, and spectral roughness exponent αs) are approximately equal within the small temporally correlated regime, and satisfy αloc≈α<αs for the large temporally correlated regime, and the difference between αs and α increases with increasing the temporal correlation exponent θ. 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.

Suggested Citation

  • Hu, Xiongpeng & Hao, Dapeng & Xia, Hui, 2023. "Improved finite-difference and pseudospectral schemes for the Kardar–Parisi–Zhang equation with long-range temporal correlations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 619(C).
    • Handle: RePEc:eee:phsmap:v:619:y:2023:i:c:s0378437123002996
    DOI: 10.1016/j.physa.2023.128744
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    References listed on IDEAS

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    1. Takeuchi, Kazumasa A., 2018. "An appetizer to modern developments on the Kardar–Parisi–Zhang universality class," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 504(C), pages 77-105.
    2. Rafael Gallego & Mario Castro & Juan M. López, 2016. "On the origin of multiscaling in stochastic-field models of surface growth," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 89(9), pages 1-7, September.
    3. Katzav, Eytan, 2013. "Fixing the fixed-point system—Applying Dynamic Renormalization Group to systems with long-range interactions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(8), pages 1750-1755.
    4. Verma, Mahendra K., 2000. "Intermittency exponents and energy spectrum of the Burgers and KPZ equations with correlated noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 277(3), pages 359-388.
    5. Moser, Keye & Kertész, János & Wolf, Dietrich E., 1991. "Numerical solution of the Kardar-Parisi-Zhang equation in one, two and three dimensions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 178(2), pages 215-226.
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