A typical stochastic partial differential equation (SPDE) used in surface growth modeling is the Kardar-Parisi-Zhang (KPZ) equation. This nonlinear SPDE characterizes the evolution of a growing interface influenced by random noise, balancing deposition, surface relaxation, and stochastic effects.

🌵MATLAB snippet

Stochastic Partial Differential Equations (SPDEs) are often used in modeling surface growth phenomena. One of the most common SPDEs for this purpose is the Kardar-Parisi-Zhang (KPZ) equation. The KPZ equation models surface growth where the height ( h(x, t) ) evolves over time due to random fluctuations, and is given by:

$$ \frac{\partial h(x, t)}{\partial t} = \nu \nabla^2 h(x, t) + \frac{\lambda}{2} (\nabla h(x, t))^2 + \eta(x, t) $$

where:

• ( $h(x, t)$ ) is the height of the surface at position ( $x$ ) and time ( $t$ ),
• ( $\nu$ ) is a positive constant representing the surface tension,
• ( $\lambda$ ) controls the strength of the non-linear term,
• ( $\eta(x, t)$ ) is a noise term, typically modeled as Gaussian white noise with zero mean.

MATLAB Implementation Overview

To simulate this SPDE, you can discretize the equation using finite difference methods for the spatial and temporal derivatives. Here's a general outline of how to set up a simulation in MATLAB:

1. Define Parameters:
   • Spatial grid ( $dx$ ), temporal step ( $dt$ ),
   • Parameters ( $\nu$ ), ( $\lambda$ ),
   • Noise strength for ( $\eta(x, t)$ ).
2. Initialize the Surface:
   • Initialize ( $h(x, 0)$ ) as a flat surface or with random initial conditions.
3. Iterate Over Time:
   • Use explicit or semi-implicit methods to iterate the solution over time.

Sample MATLAB Code

Below is a simplified version of MATLAB code to simulate the KPZ equation:

 % Parameters
 L = 100; % Length of the domain
 N = 100; % Number of grid points
 dx = L / (N - 1); % Spatial step size
 dt = 0.01; % Time step size
 T = 10; % Total simulation time
 nu = 1.0; % Diffusion coefficient
 lambda = 0.5; % Non-linear term coefficient
 noise_strength = 0.1; % Strength of the noise
 ​
 % Spatial grid
 x = linspace(0, L, N);
 h = zeros(1, N); % Initial height profile
 ​
 % Time loop
 num_steps = floor(T / dt);
 for t = 1:num_steps
     % Compute Laplacian of h
     Laplacian_h = (circshift(h, [0, -1]) - 2 * h + circshift(h, [0, 1])) / dx^2;

     % Compute gradient squared term
     grad_h = (circshift(h, [0, -1]) - circshift(h, [0, 1])) / (2 * dx);
     grad_sq = grad_h.^2;

     % Add noise term
     eta = noise_strength * randn(1, N);

     % Update the surface
     h = h + dt * (nu * Laplacian_h + (lambda / 2) * grad_sq + eta);
 end
 ​
 % Plot the result
 plot(x, h);
 title('Simulated Surface Growth');
 xlabel('Position x');
 ylabel('Height h(x)');

Explanation:

• The Laplacian is computed using a finite difference approximation.
• The gradient term ( $(\nabla h(x))^2$ ) is evaluated using central differences.