I want to discretize the following reaction diffusion system:
$\frac{\partial u(x,y,t)}{dt}=\nabla ^2u+ u(1-u)-\frac{uv}{u+\alpha v}$,
$\frac{\partial v(x,y,t)}{dt}=d\nabla ^2v+ \delta v\left(\beta-\frac{v}{u}\right)$.
$\alpha,\beta,\delta,d$ are positive parameters.
Question 1: The numerical integration of the above system are to be performed by (i) using non-standard finite difference scheme for the reaction part and (ii) five point explicit finite difference scheme for the diffusion part.
Question 2: It is also asked to use five point finite difference schemes for the diffusion part and the fourth order Runge–Kutta method for the reaction part?
Answer of question 1: Please correct my non-standard finite difference scheme (discretization technique) given below:
$\frac{u^{n+1}_{i,j}-u^n_{i,j}}{\Delta t}= \frac{u^{n}_{i+1,j}+u^n_{i-1,j}+u^n_{i,j+1}+u^n_{i,j-1}-4u^n_{i,j}}{h^2}+ \color{red}{u^n_{i,j}(1-u^{n+1}_{i,j})-\frac{u^n_{i,j}v^n_{i,j}}{u^n_{i,j}+\alpha v^n_{i,j}}}$,
$\frac{v^{n+1}_{i,j}-v^n_{i,j}}{\Delta t}=d\frac{v^{n}_{i+1,j}+v^n_{i-1,j}+v^n_{i,j+1}+v^n_{i,j-1}-4v^n_{i,j}}{h^2}+ \color{red}{\delta v^{n}_{i,j}\left(\beta-\frac{v^{n+1}_{i,j}}{u^{n}_{i,j}}\right)}$.
Query: Please specially check the red part. Is there any other, better way to answer the question? Please correct, if there is any error.
Answer of question 2: I have used the five point finite difference scheme for the diffusion part. I failed to write the fourth order Runge–Kutta method for the reaction part. Please modify the non-standard finite difference scheme for Question 2.
I would like to request you to answer both the questions in details. Thanks a lot for your support. It is very essential for me to learn the technique of non-standard finite difference.