Is it valid to approximate $u_t$ for a second order central finite difference?.
I mean $u_t=\frac{u_m^{n-1}-2u_m^{n}+u_m^{n+1}}{k^2}$, does it approximate the first partial differential derivate of $u$ with respecto of $t$?.
I have the next PDE system and I want to use finite difference, but is it correct to approximate the first partial difference of $u$ with respect of $t$ for a second order central difference scheme in time?. For what I've been stuying it will be correct if and only if $u_{tt}=\frac{u_m^{n-1}-2u_m^{n}+u_m^{n+1}}{k^2}$, I mean, the second derivate of $u$ with respect of $t$.