Solution of the 1D KPZ Equation by Explicit Methods

Abstract

The Kardar–Parisi-Zhang (KPZ) equation is examined using the recently published leapfrog–hopscotch (LH) method as well as the most standard forward time centered space (FTCS) scheme and the Heun method. The methods are verified by reproducing an analytical solution. The performance of each method is then compared by calculating the average and the maximum differences among the results and displaying the runtimes. Numerical tests show that due to the special symmetry in the time–space discretisation, the new LH method clearly outperforms the other two methods. In addition, we discuss the effect of different parameters on the solutions.

1. Introduction

One of the successful equations for describing a class of dynamic nonlinear phenomena is the Kardar–Parisi–Zhang (KPZ) equation [1]. The application of this equation varies widely in topics such as vapour deposition, directed polymers, bacterial colony growth and superconductors [2,3]. There are a number of computational studies related to discrete model simulations such as the Eden model, ballistic deposition models [4] and directed polymer model. All of these provide important features in the physical processes through simulation efficiency. The introduction of direct numerical integration is also an important point that requires more intensive computations. The first large scale numerical integration of the KPZ equation was performed by Amar and Family and the discrete Gaussian model was verified in [5]. Later, Moser improved his accuracy with further works [6,7]. However, the KPZ equation is not just a nonlinear equation that is applied by a similar method. To verify the theoretical predictions, numerical and analytical investigations are performed for the KPZ equation with correlated noise [8] and with quenched noise in anisotropic media [9], or in reaction–diffusion systems with multiplicative noise [10], for the Kuramato–Shivashisky equation of flame front propagation [11,12] and for the epitaxial growth equation [13].

In general, the direct approach to studying the growth equation is numerical integration and it can be seen as the ideal form of the equation that allows us to fully control the investigation. Unfortunately, Newman and Bray [14] reported some disadvantageous properties of the conventional numerical integration scheme, such as instability and an unphysical fixed point. Later works [15,16] reported that during numerical simulation instability can occur even in the case of small time steps. Lam and Shin [17] found that direct numerical integration by conventional finite difference schemes actually do not approximate the continuum KPZ equation. Previously, Amar and Family [18] integrated a similar equation using a generalized nonlinear term. The scaling behaviour of the KPZ equation was in most cases found to be different from the continuum equation. They even explained the results of their studies on KPZ nonlinear terms combining the effects of noise and nonlinearity.

In the last 30 years, different kinds of numerical methods have been proposed for the KPZ equation. These were implemented with various discretizing formulas of the nonlinear term. However, the diffusion term was mostly handled by the most standard forward time centred space (FTCS), where the time discretisation is based on the explicit Euler method.

The application of the discrete variational formulation to the KPZ equation has been discussed. An alternative approach to other well-known techniques, the variational analytical solutions of KPZ were introduced by Wio et al. in [19,20,21] and non-equilibrium potential has been obtained to understand radial growth on a growing domain. In [19], a relation between the real-space discretisation schemes was examined. The authors provided discrete schemes of the KPZ equation, and they discussed the role of the Galilean invariance for discrete representations. In [21], the properties of a functional related to the KPZ equation are investigated. The main result is a path integral scheme; and, the authors defined expressions for the probability of entropy production along a trajectory and they obtained integral fluctuation theorems.

The thermodynamic uncertainty relation for the (1 + 1) dimensional Kardar–Parisi–Zhang equation on a finite spatial interval was considered by Niggeman and Seifert [22]. Numerical simulations compared with theoretical predictions showed convincing agreement.

In the paper [23], the author reviewed KPZ class growth models to investigate roughness scaling using Cayley trees. Height fluctuations have been shown to be a consequence of boundary effects.

Cartes et al. [24] studied the analytical laws of the scale-dependent correlation time to follow the expected crossover from the short-distance Edwards–Wilkinson scaling to the universal long-distance Kardar–Parisi–Zhang scaling.

In a recent paper Gomes-Filho et al. [25] gave accurate results on growth exponents and on the probability of height distributions in 2 + 1 dimensions.

In this work, we examine not only conventional direct numerical integration approaches for the KPZ equation in 1 + 1 dimensions, but also adapt a recently published stable leapfrog–hopscotch (LH) algorithm [26]. Our aim is to propose a new approach for the numerical solution of the KPZ equation with noise terms that provide an easier way to overcome the difficulties of the conventional schemes. After validation using an analytical solution, and comparing the properties of the methods, we study the impact of different values of the parameters and two different noise terms.

2. The Previously Used Methods and the Newly Proposed Method

2.1. The KPZ Equation

The KPZ equation presents the local growth rate of a profile $h\left(x,t\right)$ at a substrate position x and time t [1]:

$$\frac{\partial h\left(x,t\right)}{\partial t}=\upsilon {\nabla }^{2}h+\frac{\lambda }{2}{\left(\nabla h\right)}^2+k\left(x,t\right)$$

(1)

where υ and λ are the diffusion coefficient and the nonlinear parameter, respectively. In this paper, we are going to solve the KPZ equation not only with different parameters υ and λ, but with two different noise terms $k\left(x,t\right)$, namely with Brownian and with Gaussian noise. The effect of these noise terms will be discussed in more detail in Section 3.4.

In our research, we apply traditional and new numerical methods to the time-integration of the spatially discretised KPZ equation Our goal is to find stable and efficient discretisation methods for the KPZ equation and to investigate difference or similarities from the previously proposed traditional method. Therefore, we use each method with different values of linear υ and nonlinear λ terms.

2.2. Forward Time Centered Space Scheme Adapted to KPZ Equation

In the paper [6], Moser et al. introduced spatial derivatives of the right-hand side of the KPZ Equation (1). It was discretised using standard forward–backward differences on a cubic (2) grid with lattice constant $\Delta x$, which is also known as a forward time centred space (FTCS) [27].

$$h_i^{n+1}=h_i^n+r\left(h_{i+1}^n+h_{i-1}^n-2h_i^n\right)+\mu {\left(h_{i+1}^n-h_{i-1}^n\right)}^2+\Delta t k\left(x,t\right)$$

(2)

where Δt is the step size and $t_{i+1}=t_i+\Delta t$, $r=\frac{\upsilon \Delta t}{\Delta x^2}$ and $\mu =\frac{\lambda \Delta t}{8\Delta x^2}$ are the appropriate mesh ratios. We use a for loop going through the nodes to calculate the right-hand side of Equation (2) and omit the old values. We use only one array for the variable h, which has as many elements as the number of nodes. However, when we calculate $h_i^{n+1}$, the value of $h_{i-1}^n$ is still necessary, thus we have to introduce an auxiliary temporary array variable to store the calculated values, and only after the completion of the loop can we load the new values $h_i^{n+1}$ to the array. Therefore, with its speed, the seemingly simplest algorithm can still be surpassed, as we will show later.

2.3. Heun’s Method

The Heun method is an improved or a modified Euler’s method applied in computational science and mathematics. It represents the explicit trapezoidal rule [26,28], which is a two-stage Runge–Kutta method. This method was originally proposed to solve ordinary differential equations (ODEs) with given initial conditions:

$$y^{\prime }\left(t\right)=f\left(t,y\left(t\right)\right), t\left(t_0\right)=y_0$$

(3)

The procedure in this case is the following. At the first stage, Heun’s method calculates the intermediate value $y^{pred}$ and then the final approximation $y^{n+1}$ at the next integration point:

$$\begin{gathered} y^{pred}=y^n+hf\left(t^n,y^n\right) \\ {y}^{n+1}=y^n+\frac{h}{2}\left[f\left(t^n,y^n\right)+f\left(t^{n+1},y^{pred}\right)\right] \end{gathered}$$

(4)

Although the rate of convergence of Heun’s method is two, thus it is usually more accurate than the simple explicit (Euler) method, its stability (CFL) limit is unfortunately the same [29,30].

The predictor–corrector type that Heun’s method applied to the KPZ equation reads as:

$$\begin{gathered} h_j^{pred}=h_j^n+r\left(h_{j-1}^n+h_{j+1}^n-2h_j^n\right)+\mu {\left(h_{j+1}^n-h_{j-1}^n\right)}^2+k\left(x,t^n+\Delta t/2\right)\Delta t \\ {h}_i^{n+1}=h_i^n+\frac{r}{2}\left(h_{i-1}^n+h_{i+1}^n-2h_i^n+h_{j-1}^{pred}+h_{j+1}^{pred}-2h_j^{pred}\right)+\mu \left({\left(h_{i+1}^n-h_{i-1}^n\right)}^2+{\left(h_{j+1}^{pred}-h_{j-1}^{pred}\right)}^2\right)+k\left(x,t^n+\frac{\Delta t}{2}\right)\Delta t. \end{gathered}$$

(5)

When Heun’s method is implemented by two for loops, we need not only one extra array to store $h_j^{pred}$, but a temporary array as in the FTCs method. This makes a time step of Heun’s algorithm slower and memory-consuming, although, to a much lesser extent than in the case of implicit methods.

2.4. Leapfrog–Hopscotch Method

The leapfrog–hopscotch structure was first proposed and explained in our recent paper [26]. Similar to the original odd–even hopscotch (OEH) algorithm introduced five decades ago by Gordon [31] and Gourlay [32], one must divide the grid into two subgrids of odd and even nodes (light and dark blue dots in Figure 1, respectively) such that the nearest neighbours of odd nodes are always even and vice versa. The calculation starts with a half-sized time step for the odd nodes using the initial $h_i^0$ values, symbolised by the green arrows in Figure 1. Then, full time steps are made to calculate alternately the even and the odd nodes (light and dark blue arrows, respectively) until one reaches the final time, where the time step size must also be halved for odd nodes (orange arrows). One can see that in each step, the latest available u values of the neighbours (denoted by $h_{i\pm 1}^{\mathrm{recent}}$) are used, thus the method is explicit. We mention that this LH structure was thoroughly examined for the diffusion equation [26]. According to a large number of numerical experiments, the UPFD formula is optimal for use in the zeroth time step and the symmetric $\theta =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ formula in all other steps, thus we will apply only these formulas and we will call this concrete method (the LH time–space structure and the formulas) “the LH method”. Due to the special symmetry of the time–space discretisation and the $\theta =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, this method has excellent properties, as we will see in the following.

We express the new value of the h variable in the following form in the case of the one space-dimensional KPZ equation [33,34] at the zeroth step:

$$h_i^{n+1}=\frac{2h_i^n+r\left(h_{i-1}^{recent}+h_{i+1}^{recent}\right)+\mu {\left(h_{i-1}^{recent}-h_{i+1}^{recent}\right)}^2+k\left(x,t^n+\Delta t/2\right)\Delta t}{2\left(1+r\right)}$$

(6)

and at all other steps (denoted by 1 and 2 in Figure 1):

$$h_i^{n+1}=\frac{\left(1-r\right)h_i^n+r\left(h_{i-1}^{recent}+h_{i+1}^{recent}\right)+\mu {\left(h_{i-1}^{recent}-h_{i+1}^{recent}\right)}^2+k\left(x,t^n+\Delta t/2\right)\Delta t}{1+r}$$

(7)

except the last, which is a half time step thus the substitution $\Delta t\to \raisebox{1ex}{$\Delta t$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,r\to \raisebox{1ex}{$r$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,\mu \to \raisebox{1ex}{$\mu $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ must be performed in (7). We note that if µ = 0 and k = 0 then one obtains back the original LH method’s form developed for the diffusion equation. One can see that due to the time–space structure, the LH method does not use any extra arrays for the temporary values of h, and that is why it has less memory requirement, and it can be slightly faster than the simplest FTCS method.

2.5. Stability Considerations

Since the conventional explicit methods have severe stability problems, usually implicit methods are used to solve the linear diffusion equation and most of its nonlinear modifications. For example, the explicit Euler (FTCS) is stable for the linear diffusion equation with coefficient υ only if the time step size is small enough, i.e., $\frac{\nu \Delta t}{\Delta x^2}\le \frac{1}{2}$ (this is also called the CFL limit). However, implicit methods not only require more computational effort (especially in the nonlinear case), but they are very slow and memory consuming in the case of a large number of nodes/cells and if the matrix is not tridiagonal, which is always true in 2 or more space dimensions.

In [26], we analytically proved and numerically demonstrated that the new LH method is unconditionally stable for the linear diffusion equation, and it has roughly the same accuracy as Heun’s method. The LH method is almost as accurate for stiff systems as the implicit Crank–Nicolson scheme, but it is much faster. This encouraged us to adapt the LH method to the KPZ equation.

However, the KPZ equation contains a nonlinear term with coupling coefficient λ and a noise term, besides the linear diffusion term. Dasgupta et al. showed [16] that in the discretised model, isolated pillars or grooves are prone to increase indefinitely if their height exceeds a critical value, which depends on a single parameter proportional to $\lambda \sqrt{D/{\nu }^3}$, where D is the amplitude of the noise term. They argued that this instability is inherent in the numerical treatment due to the space discretisation of the continuous equation. Our observations reinforce these findings in the case of the LH method as well, but the systematic examination of the probability of the appearance of these instabilities is out of the scope of this paper.

What we stress here is that we are talking about two qualitatively different kinds of instabilities. In the original continuous equation, the nonlinear term yields the growth of pillars and grooves, while the diffusion tries to smooth them out. However, in the case of the conventional explicit schemes, the diffusion itself also causes instability. The LH method is close to being ideal from this point of view as well: stronger diffusion means faster smoothing and thus increased stability.

3. Numerical Results

3.1. Verification Using an Analitical Solution

If we introduce the variable $\omega =x+ct$, then Equation (1) with the Brownian noise term can be written as follows:

$$\frac{\partial h\left(x,t\right)}{\partial t}=\upsilon {\nabla }^{2}h+\frac{\lambda }{2}{\left(\nabla h\right)}^2+\frac{a}{{\omega }^2 }$$

(8)

For verification, we reproduce the following recently published analytical solution to Equation (8) [35]:

$$h^{\mathrm{exact}}\left(\omega \right)=\frac{1}{\lambda }\left\{c\omega +\nu \ln \left(\frac{{\lambda }^2{I}_d{\left(\phi \right)}^2}{c^2\omega {\left[K_d\left(\phi \right)I_{d-1}\left(\phi \right)+I_d\left(\phi \right)K_{d+\frac{1}{2}}\left(\phi \right)\right]}^2}\right)\right\}$$

(9)

where a, c = const., $\phi =\frac{c\omega }{2\nu }$, while $I_d\left(\phi \right)$ and $K_d\left(\phi \right)$ are the modified Bessel functions of the first and the second kind with the subscript of $d=\frac{\sqrt{{\nu }^2-2a\lambda }}{2\nu }$.

The following parameters are used in the numerical calculation:

$$\nu =1,\lambda =6,a=0.05,c=1,x\in \left[0,6\right],t\in \left[1,2\right]$$

(10)

The number of space nodes is 300, thus the space step size is Δx = 0.02.

For the initial and the boundary conditions, the analytical solution is used at t = 1 as the initial condition and the Dirichlet boundary conditions are taken at the edges of the spatial interval. We note that the evaluation of the modified Bessel functions to obtain the initial and the boundary conditions was by far the most time consuming part of the code.

The (global) numerical error function is the absolute difference of the numerical solutions $u_j^{\mathrm{num}}$ produced by the examined method and the analytical solution $h_j^{\mathrm{exact}}=h^{\mathrm{exact}}\left(x=x_j\right)$ at the final time. We use these errors of the nodes to calculate the maximum error:

$$\mathrm{Error}\left(L_{\infty }\right)=\underset{1\le j\le N}{\max }\left|u_j^{\mathrm{exact}}\left(t=2\right)-u_j^{\mathrm{num}}\left(t=2\right)\right|$$

(11)

The $L_{\infty }$ errors as a function of the time step size h can be seen in Figure 2.

One can see that the LH method reaches the minimum error (determined by space discretisation) at about 2 × 10⁻³, while the FTCS and the Heun methods are unstable above 2 × 10⁻⁴. It means that the LH method can be safely used with an order of magnitude larger time step sizes than the examined conventional methods.

3.2. Comparison of Methods

We continue the investigation in those cases where the analytical solution is not known. The spatial length of the simulated system is $L=32$, so $x\in \left[0, 32\right]$, while the space step size is Δx = 0.01, thus the number of nodes is $Nx=3201$. In order to minimise the effect of the boundaries, we wanted to use periodic boundary conditions. However, the noise terms are not periodic functions of space, therefore we constructed a special modification of the periodic boundary conditions, when the previous (left) and the following (right) copy of the system is not only shifted but mirrored as well. It is implemented by taking $h_1=h_3$ and $h_{N_x}=h_{N_x-2}$ in the case of each method.

The initial time of the simulation is t_in = 0.01 (note that the term $x+ct$ is present in the denominator of the Brownian noise term, thus x and t cannot be zero at the same time), and the final time is t_fin = 0.1, 1 or 10. The time step size is chosen as ${10}^{-5}$ for the FTCS and the Heun methods, but it is ${10}^{-5}$ and ${10}^{-4}$ for the LH method. In this section we fix the parameters as follows:

$$\upsilon =1,\lambda =6,a=1,c=1$$

(12)

In our previous research works [36,37], we used different initial conditions with different amplitudes for Equation (1). Here, all numerical simulations for each scheme were performed with the same initial condition $h\left(x,0\right).$

$$h\left(x,t=0\right)=\sin \left(\pi ·\frac{x}{4}\right)+\cos \left(\pi ·\frac{x}{4}\right)$$

(13)

For the simulations to measure the running times, we used a laptop computer (ASUS, Taiwan) with a 2.6 GHz Intel I CITM i7-10750H CPU, 8.0 GB RAM with the MATLAB R2020b software(The MathWorks, Inc., Portola Valley, CA, USA) in which there was a built in tic-toc function to measure the total running time of the tested algorithms.

For all three methods, with the parameters given in Equation (12), the simulations were performed and the results are presented in Figure 3.

One can see in Figure 3 that the curves are indistinguishable, which means that each method is accurate. The (global) numerical difference is the absolute difference of the numerical solutions $h_i^{num}$ produced by the examined methods at final time t_fin for the KPZ equation: the FTCS scheme, the Heun method and the LH method for 10⁻⁵ and 10⁻⁴ time step size. For brevity, we denote this latter case, the LH method with 10⁻⁴ time step size as LH*.

In order to find an individual difference in the nodes or cells, we calculate the maximum and the average differences in the following ways:

$$D_i^{\mathrm{FTCS},\mathrm{Heun}}=\left|h_i^{\mathrm{FTCS}}-h_i^{\mathrm{Heun}}\right|,D_i^{\mathrm{FTCS},\mathrm{LH}}=\left|h_i^{\mathrm{FTCS}}-h_i^{\mathrm{LH}}\right|,D_i^{\mathrm{Heun},\mathrm{LH}}=\left|h_i^{\mathrm{Heun}}-h_i^{\mathrm{LH}}\right|,$$

(14)

Average differences (L₁ errors)

$$L_1^{FTCS,Heun}={{\displaystyle \sum }}_i\frac{D_i^{FTCS,Heun}}{Nx}, L_1^{LH,LH*}={{\displaystyle \sum }}_i\frac{D_i^{LH,LH*}}{Nx},\mathrm{etc}.$$

(15)

Maximum differences (L_∞ errors)

$$L_{\infty }^{FTCS,Heun}=\underset{i}{\max }\left\{D_i^{FTCS, Heun}\right\},L_{\infty }^{LH,LH*}=\underset{i}{\max }\left\{D_i^{LH,LH*}\right\},\mathrm{etc}.$$

(16)

In

Table 1. Average difference without noise term.

tfin	${\mathit{L}}_1^{\mathit{F}\mathit{T}\mathit{C}\mathit{S},\mathit{H}\mathit{e}\mathit{u}\mathit{n}}$	${\mathit{L}}_1^{\mathit{F}\mathit{T}\mathit{C}\mathit{S},\mathit{L}\mathit{H}}$	${\mathit{L}}_1^{\mathit{H}\mathit{e}\mathit{u}\mathit{n},\mathit{L}\mathit{H}}$	${\mathit{L}}_1^{\mathit{F}\mathit{T}\mathit{C}\mathit{S},\mathit{L}\mathit{H}*}$	${\mathit{L}}_1^{\mathit{H}\mathit{e}\mathit{u}\mathit{n},\mathit{L}\mathit{H}*}$	${\mathit{L}}_1^{\mathit{L}\mathit{H},\mathit{L}\mathit{H}*}$
0.1	3.48 × 10−7	3.47 × 10−7	2.70 × 10−9	3.86 × 10−4	3.88 × 10−4	3.88 × 10−4
1	5.45 × 10−7	5.42 × 10−7	2.45 × 10−9	4.30 × 10−5	4.34 × 10−5	4.34 × 10−5
10	3.20 × 10−7	3.19 × 10−7	1.10 × 10−9	3.20 × 10−7	8.38 × 10−8	8.38 × 10−8

and

Table 2. Maximum difference without noise term.

tfin	$\underset{\mathit{i}}{\mathbf{max}}\left\{{\mathit{D}}_{\mathit{i}}^{\mathit{F}\mathit{T}\mathit{C}\mathit{S}, \mathit{H}\mathit{e}\mathit{u}\mathit{n}}\right\}$	$\underset{\mathit{i}}{\mathbf{max}}\left\{{\mathit{D}}_{\mathit{i}}^{\mathit{F}\mathit{T}\mathit{C}\mathit{S},\mathit{L}\mathit{H}}\right\}$	$\underset{\mathit{i}}{\mathbf{max}}\left\{{\mathit{D}}_{\mathit{i}}^{\mathit{H}\mathit{e}\mathit{u}\mathit{n}, \mathit{L}\mathit{H}}\right\}$	$\underset{\mathit{i}}{\mathbf{max}}\left\{{\mathit{D}}_{\mathit{i}}^{\mathit{F}\mathit{T}\mathit{C}\mathit{S}, \mathit{L}\mathit{H}*}\right\}$	$\underset{\mathit{i}}{\mathbf{max}}\left\{{\mathit{D}}_{\mathit{i}}^{\mathit{H}\mathit{e}\mathit{u}\mathit{n},\mathit{L}\mathit{H}*}\right\}$	$\underset{\mathit{i}}{\mathbf{max}}\left\{{\mathit{D}}_{\mathit{i}}^{\mathit{L}\mathit{H},\mathit{L}\mathit{H}*}\right\}$
0.1	6.91 × 10−7	6.93 × 10−7	8.56 × 10−9	6.66 × 10−4	6.66 × 10−4	6.66 × 10−4
1	1.32 × 10−6	1.31 × 10−6	6.57 × 10−9	1.47 × 10−4	1.48 × 10−4	1.48 × 10−4
10	3.24 × 10−7	3.23 × 10−7	1.12 × 10−9	4.66 × 10−7	1.52 × 10−7	1.51 × 10−7

, we show the average and the maximum differences of the methods, namely the FTCS, the Heun and the LH with two different time step sizes Δt. It is clear that due to enhanced stability, the LH method can be used with larger time step sizes for further investigation of the KPZ equation.

In addition, the total running time of the tested algorithms is presented in

Table 3. Running time differences between the methods.

Methods	Running Time (s)
	tfin = 0.1	tfin = 1.0	tfin = 10.0
FTCS method, Δt = 10−5	0.3232	2.9663	28.6288
Heun’s method, Δt = 10−5	0.6573	6.2133	73.8652
LH method, Δt = 10−5	0.3135	2.5480	24.4324
LH method, Δt = 10−4	0.0825	0.3105	2.4560

. It is shown that in various time t, the method’s running time is different and the fastest is the leapfrog–hopscotch method for any time length. It is slightly faster than the FTCs method with the same time step size, as we mentioned before.

Note that when the time step size Δt is larger than 10⁻⁵, the methods are unstable except for the LH method. This proves that our new method is not only the fastest, but also more stable even in larger time steps Δt. This numerical experiment demonstrated again that this new explicit method is very effective for the KPZ equation. We think that it will be very promising, especially in two or three space-dimensional numerical experiments since they include a large amount of the nodes or cells.

3.3. The Impact of Different Parameter Values (Without Noise Term)

In this section, we investigate the effect of different parameters, such as the coefficient υ, λ, a and t_fin. When one of the parameters are changed, all other parameters are fixed: υ = 1, λ = 6, a = 1, c = 1.

For the sake of verification, we still use not only the LH, but the FTCS and Heun’s methods as well.

Figure 4 illustrates the solutions obtained with different methods for the linear diffusion parameter υ. It is clear that increasing the strength of the diffusion decreases the waviness of the surface. We note that when υ = 6.0, the traditional methods (FTCS and Heun) are unstable, and they cannot be used even with this small time step size.

In Figure 5, we illustrate the behaviour as a function of the nonlinear term parameter λ for final time t_fin = 1. One can see that the function is shifting to higher levels when the coefficient of the nonlinear term $\lambda$ is increasing and the waviness of the function decreases.

Here, we are interested in the effect of the time on the used methods. In Figure 6, we present the numerical solution of the discretised KPZ equation for different final times. Therefore, we perform the simulation for three different final times t_fin = 0.1, 1.0 and 10 with the parameters given in Equation (12). One can conclude that the waviness of the surface is decreasing as time elapses.

After examining each method with various parameters, we have more information on how the surface formation occurs. In the simulations, each increased parameter decreases the waviness of the function. However, as we mentioned above, if the linear term parameter has a higher value ($\upsilon \ge 6)$, then only the LH method is stable, while the FTCS and Heun’s methods are not stable.

3.4. Comparison of Various Noise Term Effects

Once we verified and proved the most stable method, we used the leapfrog–hopscotch method to numerically solve the KPZ equation with different noise terms and various parameters. In our previous works, we analysed the solutions with the noise term $k\left(\omega \right)=a{\omega }^2$ [38,39]. In our current numerical simulation, we examine Gaussian noise term $a{e}^{-{\omega }^2}$ and Brownian noise term $a/{\omega }^2$ when all the physical parameters are set as in Equation (17). However, some parameters are investigated separately to show their effect on the solution and to show their effect on the system. If not explicitly stated otherwise, the values of the parameters will be the following:

$$\upsilon =1,\lambda =1,a=1,c=1, \Delta t={10}^{-5}$$

(17)

Figure 7 shows that the Brownian noise term’s effect is higher than the Gaussian noise term. In this simulation, the amplitude of the noise terms is $a=1$ in both cases. The Brownian noise function $k$ has quite a high value that is close to the origin due to the small values of x and t in the denominator, thus it remarkably raises the function h at the left side of the figure, but this effect vanishes for larger values of $x$. On the other hand, the h function in the case of the Gaussian noise is only slightly different than h without a noise term, even for $x=0$.

We increased the time t_fin of the simulation from 1 to 10. In this case, we see from Figure 8 that the effect of the Brownian noise term is again much stronger than the Gaussian noise term and the elevation on the left side requires a longer length of x to reach the non-elevated surface (right side of the figure) without a noise term. Similar to the previous result in Figure 7, the function h in case of Gaussian noise is slightly smoother and the effect of the noise is small.

As stated above, the noise term has significant effects on the system. Therefore, we performed our method with a higher amplitude than a = 1. We fixed the amplitude as a = 10.0 and it was simulated with all the noises. Figure 9 for t_fin = 1 and Figure 10 for t_fin = 10 show that when the noise term amplitude is high, the results are different. However, the effect of Brownian noise is again much stronger than Gaussian noise. This proves that noise term amplitude plays a crucial role in the simulation result.

We have already discussed the average/maximum differences without noise terms and we presented all of the obtained results in Table 1, Table 2 and Table 3. We have seen that the LH* method (Lh method with increased time step size) is very accurate for a large final time. Therefore, we now simulate surface growth by the LH* method with different noise terms in longer time t_fin = 10 to see the effect of the amplitude to the smaller time stem size. As a result, the effect of the Brownian noise is significantly stronger than that of the Gaussian, which has only a slight effect, approximately x = 0 compared to the case without the noise term. Compared to Figure 9, we see that due to the longer final time—and without the noise member functions—its height is low for Gaussian. However, the function k in the case of the Brownian noise term is even higher due to the longer simulation time t_fin = 10.

We demonstrated in Figure 4 that the effect of the increased (linear) diffusion term parameter $\upsilon$ is decreased surface waviness. Now, we present the effect of the parameters $\upsilon \mathrm{and} \lambda$ on the method when the value of the diffusion term is large, $\upsilon =3.0$ and the value of the nonlinear term is small, $\lambda =0.1$. In Figure 11, one can see that the shape of the surface is very close to a sine wave and the increased linear term parameter $\upsilon$ changes Brownian noise but not the Gaussian noise term.

In Figure 12, we fixed all other parameters as in Equation (17) but only linear and nonlinear terms are changed to a very small value of $\upsilon$ and to a large value of λ. The effect is slightly similar to that in Figure 5. However, due to the small value of υ and the large value of λ, the bottoms of the wavy surface became very narrow valleys.

4. Conclusions

We have used the most standard FTCS scheme, the well-known Heun method and the recently invented leapfrog–hopscotch method to solve the KPZ equation in one spatial dimension. We have validated all methods using an analytical solution, and we compared the performance of these methods in cases of different parameters. We found that FTCS and the Heun’s methods are usually unstable above Δt = 2 × 10⁻⁵ (the concrete threshold for stability depends on the parameters), while the LH method was stable in all of the presented numerical experiments. It can be unstable only if the nonlinear coupling term λ are much higher than the linear coefficient υ and/or there are spike-like pillars or grooves. Therefore, even if the LH method is not unconditionally stable for the KPZ equation, it is obvious that it has much better stability properties than the conventional explicit methods.

The effect of the parameters on the solution for the KPZ equation, and most importantly the coefficients in the equation, have been examined. The newly proposed LH method has been used without noise and with Gaussian and Brownian noise terms to show the effect of the noise terms for fixed parameters. In addition, we have simulated two different noise terms with different parameters such as υ linear, λ nonlinear and $a$ noise term amplitude parameters. The effect of each applied parameter has been presented and discussed.