Literature on second order finite difference schemes convergence analysis

Question

Okay I am studying Klein-Gordon type equations $$ \frac{\partial^2 u}{\partial t^2} - \Delta u = f(u) $$ With initial conditions $u(0,\mathbf{x})$, $\dot{u}(0,\mathbf{x})$. And I have the following finite difference scheme (one dimensional for illustration): $$ \begin{align} v_m^{n+1} &= v_m^n + \left(\frac{u_{m-1}^n - 2 u_m^n + u_{m+1}^n}{h^2}+ f(u_m^n) \right)k, \\ u_m^{k+1} &= u_m^n + v_m^nk. \end{align} $$ I want to ask for book or text recommendations on how to perform convergence analysis. I have been following Strikwerda's book but I wanted to see if there was a more straightforward treatment when dealing with this kind of schemes.