The numerical solution of advection and wave equations, particularly nonlinear conservation laws, is still an active area of research. A more detailed treatment on finite-difference methods for scalar problems is in LeVeque LeVeque (2007). An accessible treatment of nonlinear conservation laws can be found in the monograph by LeVeque LeVeque (1992). Recent monographs aimed at hyperbolic PDEs, both scalar equations and systems, are from Trangenstein Trangenstein (2009) and LeVeque LeVeque (2002).
A wide-ranging view of computational mathematics relating to conservation laws (in many cases, nonlinear versions of the advection equation) can be found in Peter Lax’s oral history.
- LeVeque, R. J. (2007). Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. SIAM.
- LeVeque, R. J. (1992). Numerical Methods for Conservation Laws. Springer Science & Business Media.
- Trangenstein, J. A. (2009). Numerical Solution of Hyperbolic Partial Differential Equations. Cambridge University Press.
- LeVeque, R. J. (2002). Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.