There are many texts on PDEs; a fairly popular undergraduate level text is by Haberman Haberman (1998). A more advanced treatment is by Ockendon et al. Ockendon et al. (2003).
For a more traditional and analytical take on full discretization of the heat equation, one may consult Smith Smith (1985) or Morton and Mayers Morton & Mayers (2005).
Several examples of using the method of lines with spectral method approximation may be found in Trefethen’s text Trefethen (2000). A classic text on using spectral methods on the PDEs from fluid mechanics is Canuto et al. Canuto et al. (1988). The literature on the complex computational fluid dynamics is vast, but one comprehensive monograph is by Roache 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.