# Roughening in Nonlinear Surface Growth Model
## 3 Numerical Results  
In this paper, the solution to (5) is examined with numerical simulations. These results help us to explain the phenomena observed by experiments and validate the mathematical model. So, we follow the stress effects in understanding the physical phenomena. Our choice for the initial condition applied to (5) which mimics the initial surface structure of the substrate:  
\[h(0,x)=1 cos\frac{x}{16}\Big{(}1+sin\frac{x}{16}\Big{)} \tag{6}\]  
The solution of the one-dimensional combined CKS Equation (5) subjected to (6) is determined numerically by using Fourier spectral collocation in space and the fourth order Runge-Kutta exponential time differencing scheme for time discretization. Numerical solution for height profiles are calculated for different values of \(r\) with the following data:  
\[x\in[0,32\pi],\ t\in[0,250],N=256,\Delta t=1/100.\]  
The surface height profile \(h(t,x)\) is shown on Figure 1 calculated from the nonlinear deterministic growth Equation (5) in 1+1 dimension using the parameter \(r=0.01\). We can observe that the evaluation of \(h\) starts about \(t=\)50 for initial condition (6). We note that for taking an amplitude greater than \(A=1\) in (6), the evaluation of \(h\) starts earlier. In Figure 1 we see the chaotic evaluation of the ripples which merge and split at different t and x. Figure 2 illustrates the cross-section of the surface structure at fixed time \(t=1000\) for different values of \(r\) (\(r=0.01,\ 0.5,\ 1\).) It can be seen that with increasing \(r\) the cross section became more regular. For large \(r\) (\(r=1\)), the cross section at \(t=1000\) shows periodicity like \(\left|\sin x\right|\).  
Figure 1: The height profile \(h(t,x)\) from solution of Equation (5) with \(r=0.01\).  
Figure 2: The height profile for time \(t=1000\) and for different values of parameter \(r\) (**a**) \(r=0.01\) (**b**) \(r=0.5\) (**c**) \(r=10\).  
The temporal evaluation of the film roughness described by its height profile \(h(t,x)\) can be seen in Figure 2. The surface roughness is characterized by the surface width \(w(t)\) which is defined by the rms fluctuation in the \(h(t,x)\), in time \(t\) for a linear size \(L\) of the sample.  
Statistical evaluation of the parameters is calculated on the surface (mean, standard deviation, etc.). On the profile, where data points are joined by line segments the linear interpolation is applied, but a spline interpolation provides usually closer result to the continuous profile. The texture parameters are calculated, the approximation method by summations is replaced by integrals. To ensure the correctness, the parameter definitions are given for the continuous functions, i.e., parameters are expressed with integrals.  
In modern specification standards, parameter definitions are always given for the continuous case, i.e., expressed with integrals. In case of statistical evaluation of the parameters for the microgeometry of the surface, it is convenient to centre the height function in the evaluation of the surface texture parameter, i.e., the mean height is subtracted from the height. The use of continuous definitions ensures the correctness of the definition and does not imply any numerical approximation.  
In this paper, the change of roughness in time is illustrated by the function \(w(t)\) with the deviation from the average height as follows:  
\[w^{2}(t)=\frac{1}{L}\int_{0}^{L}\bigl{[}h(x,t)-\overline{h}(t)\bigr{]}^{2}dx \tag{7}\]  
where the average height function \(\overline{h}(t)\) is calculated with the form:  
\[\overline{h}(t)=\frac{1}{L}\int_{0}^{L}h(x,t)dx.\]  
Function \(w(t)\) is used to call the mean interface width. We recall that in our calculations \(L=32\pi\).