Anomaly in Numerical Integrations of the KPZ Equation and Improved Discretization

We demonstrate and explain that conventional finite difference schemes for direct numerical integration do not approximate the continuum Kardar-ParisiZhang (KPZ) equation due to microscopic roughness. The effective diffusion coefficient is found to be inconsistent with the nominal one. We propose a novel discretization in 1+1 dimensions which does not suffer from this deficiency and elucidates the reliability and limitations of direct integration approaches. PACS numbers: 64.60.Ht, 05.40.+j, 05.70.Ln, 64.60.AK Typeset using REVTEX 1 The Kardar-Parisi-Zhang (KPZ) equation has been very successful in describing a class of dynamical self-affine interfaces [1]. Numerous simulations on discrete models for vapor deposition, bacterial colony growth, directed polymers, etc. show agreements with KPZ predictions. Being the simplest nonlinear stochastic evolution equation for interfaces, the KPZ equation is believed to be relevant to a large diversity of phenomena although experimental verifications has been controversial [1]. Many numerical investigations on the subject have concentrated on discrete models. This work focuses on another important approach, namely, direct numerical integration of the KPZ equation. Amar and Family first conducted such large-scale integrations [2]. They found scaling exponents of the resulting interfaces in agreement with those from discrete models. This conclusion is supported subsequently by more accurate works indicating the validity of the KPZ approach [3,4]. However, it has been observed that the discretized equations in the numerical integration of the KPZ equation admit peculiar properties not fully compatible with their continuum counterparts [3,5,6]. By applying Lam and Sander’s inverse method [7], we will give a quantitative demonstration and theoretical explanation of an abnormal behavior of the diffusion coefficient. We propose a novel discretization for the numerical integration of KPZ interfaces in 1+1 dimensions. Our discrete equations behaves in a much more predictable way as proved by exact solution of its steady state properties. The results should clarify the reliability and limitations of conventional numerical integration techniques on the KPZ equation. In addition, conventional direct numerical integration schemes for the KPZ equation are inefficient and numerically rather unstable at high nonlinearity [2–4]. We will give a quantitative evaluation of this instability. In contrast, the new discrete equations can be integrated substantially more efficiently with much improved stability. The KPZ equation gives the local rate of growth of the coarse-grained height profile h(x, t) of an interface at substrate position x and time t [1]: ∂h ∂t = c+ ν∇h+ λ 2 (∇h) + η(x, t), (1) where c, ν and λ are the average growth rate, the diffusion coefficient and the nonlinear 2 parameter respectively. There is an implicit lower wavelength cutoff below which h is smooth. The noise η has a Gaussian distribution and mean 0 and a correlator < η(x, t)η(x, t) >= 2Dδ(x−x)δ(t− t). Most previous works on the numerical integration of the KPZ equation adopt the finite difference and Euler’s method with the following equation [2–4]: h i = h n i +∆t[ν0(h n i+1 + h n i−1 − 2h n i ) +(λ0/8)(h n i+1 − h n i−1) ] + √