one could use linear approximation $g_{ab}=\eta_{ab}+h_{ab}$ to get linear ricci flow equation (2). How to do that? is any process ? I am studying general relativity , i just use the linear approximation : $g_{ab}=\eta_{ab}+\gamma_{ab}$, and calculate the christoffl symbol : $\Gamma_{a b}^{(1) c}=\frac{1}{2} \eta^{c d}\left(\partial_{a} \gamma_{b d}+\partial_{b} \gamma_{a d}-\partial_{d} \gamma_{a b}\right)$,finally, get the Ricci tensor:$R_{a b}^{(1)}=\partial^{c} \partial_{(a} \gamma_{b) c}-\frac{1}{2} \partial^{c} \partial_{c} \gamma_{a b}-\frac{1}{2} \partial_{a} \partial_{b} \gamma$ but there are two extra term :$\partial^{c} \partial_{(a} \gamma_{b) c}$ and $-\frac{1}{2} \partial_{a} \partial_{b} \gamma$, which not match the linearization ricci flow equation(2). what is wrong with me?
You are correct, the paper is mistaken. In fact it's well known that the linearization of the Ricci flow is not strongly parabolic, so it is impossible that the linearization should be the standard heat equation.
It looks like that article ("Modified Ricci flow and asymptotically non-flat spaces") also has some other errors or typos, such as its equation (3) which is an erroneous citation of the paper by Samuel and Roy Chowdhury.
See section 2 of Anderson and Chow "A pinching estimate for solutions of the linearized Ricci flow system on 3-manifolds" for a proper (brief) discussion of linearized Ricci flow.