Team:Valencia UPV/prueba

Diffusion

The diffusion equation is a partial differential equation which describes density dynamics in a material undergoing diffusion. It is also used to describe processes exhibiting diffusive-like behavior, like in our case.

The equation is usually written as:

where ϕ(r, t) is the density of the diffusing material at location r and time t and D(ϕ, r) is the collective diffusion coefficient for density ϕ at location r; and ∇ represents the vector differential operator del.

If the diffusion coefficient does not depend on the density then the equation is linear and D is constant.

Thus, the equation reduces to the following linear differential equation:

also called the heat equation.

Making use of this equation we can write pheromones chemicals diffusion equation with no wind effect consideration as:

Where C is the pheromone concentration, ∇ is the Laplacian operator, and D is the pheromone diffusion constant in air.

If we consider the wind, we face a diffusion system with drift and an advection term is added to the equation above.

v is the average velocity that the quantity is moving. Thus, v would be the velocity of the air flow.

For simplicity, we are not going to consider the third dimension. In 2D the equation would be:

For determining a numeric solution for this partial differential equation are used the so-called finite difference methods. The technic consists in approximating differential ratios as h is closer to zero, so are useful to approximate differential equations.

With finite difference methods, partial differential equations are replaced by its approximations in finite differences, resulting in an algebraic equations system. The algebraic equations system is solved in each node (xi,yj,tk). These discrete values describe the temporal and spatial distribution of the unknown function.

Although implicit methods are unconditionally stable so time steps could be larger and make the calculus process faster, the tool we have used to solve our heat equation is the Euler explicit method.

Although implicit methods are unconditionally stable so time steps could be larger and make the calculus process faster, the tool we have used to solve our heat equation is the Euler explicit method.

JUSTIFICATION

Euler explicit method is the simplest option to approximate spatial derivatives, in which all values are assumed at the beginning of Time.

The equation gives the new value of the pheromone level in terms of initial values in that node and its immediate neighbors. Since all these values are known the process is called explicit.

Now applying this method for the first case (with no wind consideration) we followed the next steps.

1. Split time t into n slices of equal length delta t
2. Considering the backward difference for the Euler explicit method implies that the expression that refers to the current pheromone level each time step is:
3. 3. And now considering the spatial dimension, it is applied central differences to the Laplace operator in 2D (assuming equal steps in x and y directions)