There are many texts on PDEs; a fairly popular undergraduate level text is Haberman (1998). A more advanced treatment is Ockendon et al. (2003).
For a more traditional and analytical take on full discretization of the heat equation, one may consult Smith (1985) or Morton & Mayers (2005).
Several examples of using the method of lines with spectral method approximation may be found in Trefethen (2000). A classic text on using spectral methods on the PDEs from fluid mechanics is Canuto et al. (1988). The literature on the complex computational fluid dynamics is vast, but one comprehensive monograph is Roache (1998).
For a first-hand account of the development of numerical methods for equations governing reservoir simulations (close to the heat equation), see the article by D. W. Peaceman, reprinted from Nash (1990). Those were early days in computing!
- Haberman, R. (1998). Elementary Applied Partial Differential Equations: With Fourier Series and Boundary Value Problems. Prentice Hall.
- Ockendon, J. R., Howison, S., Lacey, A., & Movchan, A. (2003). Applied Partial Differential Equations. Oxford University Press.
- Smith, G. D. (1985). Numerical Solution of Partial Differential Equations: Finite Difference Methods (3rd ed.). Clarendon Press.
- Morton, K. W., & Mayers, D. F. (2005). Numerical Solution of Partial Differential Equations: An Introduction (Second). Cambridge University Press.
- Trefethen, L. N. (2000). Spectral Methods in MATLAB. SIAM.
- Canuto, C., Hussaini, M. Y., Quarteroni, A., & Zang, T. A. (1988). Spectral Methods in Fluid Dynamics. Springer Berlin Heidelberg.
- Roache, P. J. (1998). Fundamentals of Computational Fluid Dynamics. Hermosa Publishers.
- Nash, S. (1990). A History of Scientific Computing. Addison-Wesley Publishing Company.