Anomaly in Numerical Integrations of the Kardar-Parisi-Zhang Equation

The Kardar-Parisi-Zhang ~KPZ! equation @1# has been very successful in describing a class of dynamic nonlinear phenomena. It is applied to a wide range of topics including vapor deposition, bacterial colony growth, directed polymers, and flux lines in superconductors @2,3#. Computational studies have mostly concentrated on simulations of discrete models such as ballistic deposition models, solid-on-solid models, Eden model, directed polymer simulations, and so on. They allow very efficient simulations by capturing only the essential features in the physical processes. Another important approach is direct numerical integration. This in general involves more intensive computations. Amar and Family first conducted large-scale numerical integrations of the KPZ equation and verified the universality with discrete models @4#. The accuracy was further improved by the subsequent works of Moser et al. @5,6#. Properties of various numerical integration approaches are still being investigated @7–10#. Similar techniques are not only applied to the KPZ equation but also to many related nonlinear phenomena such as growth with correlated noise @11# or quenched noise in anisotropic media @12#, reaction-diffusion systems with multiplicative noise @13#, Kuramato-Shivashisky equation for flame front propagation @14,15#, epitaxial growth @9,16#, etc. Numerical integration is in general considered to be a more direct approach for the investigation of growth equations. Ideally, it should allow full control on the precise form of the equation to be investigated. The parameters involved may also be chosen at will. Unfortunately, many obscure properties of the conventional numerical integration scheme are reported. For example, Newman and Bray @8# identified an unphysical fixed point and an associated instability in the deterministic version of the discrete equation used in the numerical integration of the KPZ equation. They further argue that the stochastic discrete equation and hence the conventional integration method cannot capture the strong coupling behavior of the continuum equation. In another work, Dasgupta, Das Sarma, and Kim @9# reported that numerical instability can occur even for very small time steps used in the numerical integration. They suggest that the instability is an intrinsic property and inferred that the discretized KPZ equation may have very different behavior from that of the continuum version. In an earlier work, Amar and Family @17# integrated a related equation with a generalized nonlinear term. Contrary to predictions from the continuum equation,