Chapter 24 — Further Reading

The finite-difference method for PDEs is a deep, well-charted field. These are the resources that teach it honestly — the numerics and the implementation — grouped by what you want from them.

Finite differences and the numerics of PDEs

• Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations (SIAM, 2007). The modern standard text. It develops exactly what this chapter sketched — the stencil, explicit vs. implicit stepping, and von Neumann stability analysis — with full rigour but a practical bent. If you read one book to go deeper, read this. (Tier 1.)
• John C. Strikwerda, Finite Difference Schemes and Partial Differential Equations (2nd ed., SIAM, 2004). The most careful treatment of stability specifically — von Neumann analysis, the Lax equivalence theorem (consistency + stability ⇒ convergence), and why the CFL condition is necessary. The book to read when you want to prove a scheme is stable, not just observe it. (Tier 1.)
• K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations (2nd ed., Cambridge, 2005). A compact, readable graduate introduction; its heat-equation and stability chapters map almost one-to-one onto §§24.2–24.4. (Tier 1.)
• William H. Press et al., Numerical Recipes (3rd ed., Cambridge). Chapter 20, "Partial Differential Equations," is a fast, opinionated tour of FTCS, the stability limit, and implicit methods, with an engineer's eye for what actually works. Read it for intuition and caveats, not for copy-paste code. (Tier 1.)

Fortran implementation

• Milan Curcic, Modern Fortran: Building Efficient Parallel Applications (Manning, 2020). Builds numerical solvers (including a tsunami/shallow-water simulation) in exactly the modern style of this book — field_t-like types, array operations, and a clean path to parallelism. The best companion for turning the math into idiomatic Fortran. (Tier 1.)
• Michael Metcalf, John Reid, and Malcolm Cohen, Modern Fortran Explained (Oxford). The authoritative language reference; consult it for the precise semantics of array-section assignment and pure functions that the stencil relies on. (Tier 1.)
• Stephen J. Chapman, Fortran for Scientists and Engineers (McGraw-Hill). A thorough teaching text; useful if you want more worked array and procedure examples before extending the solver. (Tier 1.)

Primary source — where the CFL condition comes from

• R. Courant, K. Friedrichs, and H. Lewy, "Über die partiellen Differenzengleichungen der mathematischen Physik," Mathematische Annalen 100 (1928), 32–74. The paper that introduced the stability condition now named for its authors — written to study existence of PDE solutions, decades before the computers it would come to govern. An English translation ("On the partial difference equations of mathematical physics") appeared in the IBM Journal of Research and Development 11 (1967). Worth seeing, at least once, where the idea began. (Tier 1.)

Online and tools

• fortran-lang.org — the community hub: the learning materials, stdlib, and fpm you will use to package the solver. (Tier 1.)
• The GCC / gfortran manual (gcc.gnu.org/onlinedocs) — the reference for the flags this chapter and the next parts lean on (-fcheck=all, -ffpe-trap, later -O3, -march=native, -fopenmp). (Tier 1.)
• The ParaView documentation (docs.paraview.org) — for Chapter 26, when your solver starts writing VTK and you want to watch the plate warm. (Tier 1.)

Suggested order

1. Re-read §24.4 of this chapter, then the stability chapter of Morton & Mayers or Numerical Recipes Ch. 20 for a second pass at the CFL idea.
2. Work through LeVeque's heat-equation chapters to see the theory in full, especially the derivation of the order of accuracy you will verify in the exercises.
3. Read Strikwerda on von Neumann analysis when you want to derive stability limits for schemes beyond FTCS (implicit, wave, advection).
4. Keep Curcic open beside your editor as you extend the solver toward the Chapter 38 capstone.