The Gray-Scott Model is a reaction-diffusion model that uses two imaginary species A and B which react. Due to the reaction both equations (for A and B) are coupled. The model can be used to investigate into numerical schemes and is a playground due to the fact that we can observe interesting reaction patterns if we change the model parameters slightly. If you are interested on that kind of problems, you can check out the following links for more details:

The equations that describe the problem are extended with the convective term and can be expressed for A and B as:

\[\frac{\partial A}{\partial t} + \nabla \bullet (\textbf{U} A)= D_a \nabla \bullet (\nabla A) - AB^2 + f(1-A) ~~~~ (1)\]

\[\frac{\partial B}{\partial t} + \nabla \bullet (\textbf{U} B) = D_b \nabla \bullet (\nabla B) + AB^2 - (f + k)B ~~~~ (2)\]

The characteristics of these two equations are:

  • In equation (1), A is linear in each term
  • In equation (2), B is non-linear in the second term on the right hand side
  • The second term of each equation are the coupled terms. Here A + 2B react to 3B; that means, A will be killed and B produced
  • The last term of both equations represent source terms:
    1. For A it is the continuous feeding rate of A (f = FEED); further more A is limited to the quantity 1
    2. For B it is a continuous killing rate of B (k = KILL); here we add f to k to be sure that the killing rate is always be higher than the feeding rate

A detailed derivation of the matrix of A and B (for implicit use) can be found in the publication section » Numerische Methoden II « (only in german).
Go to the Gray-Scott-Repository

  • Gray-Scott-1
  • Gray-Scott-2
  • GrayScottModel-Pattern2
  • GrayScottModel-Pattern4