11.6. Next steps

There are many texts on PDEs; a fairly popular undergraduate level text is by Haberman [Hab98]. A more advanced treatment is by Ockendon et al. [OHLM03].

For a more traditional and analytical take on full discretization of the heat equation, one may consult Smith [Smi85] or Morton and Mayers [MM05].

Several examples of using the method of lines with spectral method approximation may be found in Trefethen’s text [Tre00]. A classic text on using spectral methods on the PDEs from fluid mechanics is Canuto et al. [CHQZ88]. The literature on the complex computational fluid dynamics is vast, but one comprehensive monograph is by Roache [Roa98].

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 [Nas90]. Those were early days in computing!