A summary of numerical methods for time-dependent advection-dominated partial differential equations

We give a brief summary of numerical methods for time-dependent advection-dominated partial differential equations (PDEs), including first-order hyperbolic PDEs and nonstationary advection–diffusion PDEs. Mathematical models arising in porous medium fluid flow are presented to motivate these equations. It is understood that these PDEs also arise in many other important fields and that the numerical methods reviewed apply to general advection-dominated PDEs. We conduct a brief historical review of classical numerical methods, and a survey of the recent developments on the Eulerian and characteristic methods for time-dependent advection-dominated PDEs. The survey is not comprehensive due to the limitation of its length, and a large portion of the paper covers characteristic or Eulerian–Lagrangian methods.