# Gray Scott Model.

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 numerical schemes and is a playground because we can observe interesting reaction patterns if we change the model parameters slightly. If you are interested in 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:

• For A it represents a continuous feeding rate (f = FEED); furthermore, A is limited to the quantity 1
• For B it is a continuous reducing/killing rate (k = KILL); here we add the value f to the value k, to ensure that the killing rate is always 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).