There are many well-written books on this topic including:
J. C. Strikwerda, Finite difference schemes and partial differential equations, SIAM, 2004.
R. J. LeVeque, Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems, vol. 98, SIAM, 2007.
B. Gustafsson, H.-O. Kreiss, and J. Oliger, Time-Dependent Problems and Difference Methods, 2nd Edition, vol. 123, John Wiley & Sons, 2013.
Lloyd N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, unpublished text, 1996, available at https://people.maths.ox.ac.uk/trefethen/pdetext.html