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*, vol. 123, John Wiley & Sons, 2013.

L. N. Trefethen, *Finite difference and spectral methods for ordinary and partial differential equations,* Online Lecture Notes: https://people.maths.ox.ac.uk/trefethen/pdetext.html