I have the next nonlinear PDE system, and I want to apply a SECOND ORDER CENTRAL finite difference scheme in TIME and a SECOND ORDER CENTRAL finite difference scheme in SPACE:
$u_t=D_u u_{xx}-u+av+u^2v\\ v_t=D_v v_{xx}+b-av-u^2v$
Where $D_u,D_v,a,b $ are constants. How can I apply that second order finite difference schemes for the variables $u$ and $v$. Is it correct to use
$u_t=\frac{u_p^{n+1}-2u_p^n+u_p^{n-1}}{k^2}$?.
If not, how can I assure a second order approach centered in time and space?. Is there anyway to use an implicit scheme for time and space?. I think I have a explicit scheme but I'm using $u_t=\frac{u_p^{n+1}-2u_p^n+u_p^{n-1}}{k^2}$ and I don't think that is correct. Is there somebody that can help me out?
For the time integration, Crank-Nicholson is the classic choice for implicit second order accurate method.
You could use instead $$u_t \approx \frac{u^{n+1} - u^{n-1}}{2 \Delta t}.$$
The equation you give is for second derivatives $u_{tt}$ and can be used to obtain a second order accurate method for the spatial derivatives $u_{xx}, v_{xx}$.