There are many well-written books/notes 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.
Thomas, James W. Numerical partial differential equations: finite difference methods. Vol. 22. Springer Science & Business Media, 2013.
Arnold, Douglas N. Lecture notes on Numerical Analysis of Partial Differential Equations, 2012. 2014-2015 version is available at http://www.math.umn.edu/~arnold/8445/notes.pdf
Lloyd N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, unpublished text, 1996, available at https://people.maths.ox.ac.uk/trefethen/pdetext.html
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The above references are aimed at graduate students in applied mathematics and are an excellent intro to basic concepts like consistency, stability, and convergence of finite difference methods. However, they do not cover more advanced topics like Higher Order Accuracy, Singularly Perturbed PDEs, Long-Time Simulation and related to this Structure-Preserving Discretizations, Adaptive Mesh Refinment, the Method of Lines, Meshless Finite Difference Methods for irregularly shaped regions, and Monte-Carlo Methods for local solutions to high-dimensional PDEs. These more advanced methods are necessarily more specialized.
To conclude, let me just point out that there are two main ingredients to proving accuracy of a numerical method to a PDE (finite difference, volume or element):
- the quantity to be computed is sufficiently regular; and
- the approximation is stable and accurate in approximating this quantity.
Local accuracy is typically straightforward to verify with finite-difference approximations both in the interior and on the boundary of the PDE problem: it just requires doing a Taylor expansion. (If the problem has internal discontinuities, then this verification is not so straightforward.) However, deriving an estimate of the global error of the approximation requires more work. Typically this error is expressed in terms of the quantity you wish to compute evaluated at the numerical solution. Thus, in order to estimate this global error, one needs both a priori estimates on the quantity being approximated and some notion of stability of the numerical solution. Clearly, it really helps to understand as much as possible the properties of the continuous solution you wish to approximate, which emphasizes that understanding the PDE you wish to approximate is essential for the design and analysis of good schemes.
Its hard to be more specific about a topic as broad as "finite difference methods for PDEs", but I hope this helps!