Numerical integration of KPZ equation with restrictions

Abstract

In this paper, we introduce a novel integration method of Kardar-Parisi-Zhang (KPZ) equation. It has always been known that if during the discrete integration of the KPZ equation the nearest-neighbor height-difference exceeds a critical value, an instability appears and the integration diverges. One way to avoid these instabilities is to replace the KPZ nonlinear-term by a function of the same term that depends on a single adjustable parameter which is able to control pillars or grooves growing on the interface. Here, we propose a different integration method which consists of directly limiting the value taken by the KPZ nonlinearity, thereby imposing a restriction rule that is applied in each integration time-step, as if it were the growth rule of a restricted discrete model, e.g. restricted-solid-on-solid (RSOS). Taking the discrete KPZ equation with restrictions to its dimensionless version, the integration depends on three parameters: the coupling constant g, the inverse of the time-step k, and the restriction constant ε which is chosen to eliminate divergences while keeping all the properties of the continuous KPZ equation. We study in detail the conditions in the parameters' space that avoids divergences in the 1-dimensional integration and reproduce the scaling properties of the continuous KPZ with a particular parameter set. We apply the tested methodology to the d-dimensional case (d = 3, 4) with the purpose of obtaining the growth exponent β, by establishing the conditions of the coupling constant g under which we recover known values reached by other authors, in particular for the RSOS model. This method allows us to infer that d = 4 is not the critical dimension of the KPZ universality class, where the strong-coupling phase dissapears.

Anomalous Diffusion: A Basic Mechanism for the Evolution of Inhomogeneous Systems

“…Since the KPZ renormalization approach is valid only for 1 + 1 dimensions, questions about the validity of the Galilean invariance [164,165] for d > 1 and the existence of an upper critical dimension for KPZ [166,167] have been raised. For d > 1, the numerical simulation of the KPZ equation is not an easy task [164,165,[168][169][170][171], and the use of cellular automata models [149][150][151][152][153][154][155][156][157][158][159][160][172][173][174][175] has become increasingly common for growth simulations. Polynuclear growth (PNG), is a typical example of a discrete model that has received a lot of attention, and the outstanding works of Prähofer and Spohn [176] and Johansson [177] drive the way to the exact solution of the distributions of the heights fluctuations f (h, t) in the KPZ equation for 1 + 1 dimensions [134].…”

A new stable, explicit, and generic third‐order method for simulating conductive heat transfer

“…In Figure 8, we present the maximum errors as a function of the time step size for the same methods as in Figure 5. We also present the maximum and average global errors as a function of the total running FIGURE 11 The variable u as a function of the cell index i for the first row of the system (see Figure 3) consist of the first 100 cells. The black line with circle markers represents the initial function, the red continuous line is the high-precision solution while the blue dotted line with diamond markers are the values produced by our constant + constant + linear neighbor (CCL) algorithm for h = 10 −5 in 12.8 s. We note that this blue line together with the red one seems to be purple because they coincide times in Figures 9 and 10, respectively.…”

“…Moreover, sometimes they are not even defined on a connected subdomain of, for example, $$$ {\mathrm{\mathbb{R}}}^2 $$$, which can severely restrict their usefulness for the applications [5, 6]. Therefore, the construction of new numerical methods and testing them (against the standard methods as well) are very important, see for example, [7–11] and the references therein.…”

The Fractal Geometry of Growth: Fluctuation–Dissipation Theorem and Hidden Symmetry

“…For a solid, the crystalline symmetries are broken during the growth process, which creates the interface with a fractal dimension d f [53]. Although a numerical solution of KPZ equation was obtained with good precision [29] for d 1, 2, and 3, the exponents can be obtained in an easier way from cellular automata simulations. For example, the stochastic cellular automaton, etching model [4]; [7]; [8], which mimics the erosion process by an acid, has been recently proven to belong to the KPZ universality class [67].…”

“…Despite all effort, finding an analytical or even a numerical solution of the KPZ equation ( 2) is not an easy task [27,28]; [29]; [30,31]; [32], and we are still far from a satisfactory theory for the KPZ equation, which makes it one of the most difficult and exciting problems in modern mathematical physics [33][34][35][36][37][38][39][40][41], and probably one of the most important problems in non-equilibrium statistical physics. The outstanding works of Prähofer and Spohn [35] and Johansson [42] opened the possibility of an exact solution for the distributions of the height fluctuations f(h,t) in the KPZ equation for 1 + 1 dimensions (for reviews see [37][38][39][40][41][42]).…”