There are many well-written books on this topic including:

J. C. Strikwerda, [Finite difference schemes and partial differential equations][1], SIAM, 2004.

R. J. LeVeque, [Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems][2], vol. 98, SIAM, 2007.

B. Gustafsson, H.-O. Kreiss, and J. Oliger, [Time-Dependent Problems and Difference Methods][3], 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 Lloyd N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, unpublished text, 1996, available at https://people.maths.ox.ac.uk/trefethen/pdetext.html


  [1]: http://epubs.siam.org/doi/book/10.1137/1.9780898717938
  [2]: http://bookstore.siam.org/ot98/
  [3]: http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470900563,subjectCd-MA80.html