12.5. Next steps

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 [LeV07]. An accessible treatment of nonlinear conservation laws can be found in the monograph by LeVeque [LeV92]. Recent monographs aimed at hyperbolic PDEs, both scalar equations and systems, are from Trangenstein [Tra09] and LeVeque [LeV02].

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.